Elliptic curves over a general ring¶
Sage defines an elliptic curve over a ring \(R\) as a Weierstrass Model with five coefficients \([a_1,a_2,a_3,a_4,a_6]\) in \(R\) given by
Note that the (usual) scheme-theoretic definition of an elliptic curve over \(R\) would require the discriminant to be a unit in \(R\); Sage only imposes that the discriminant is nonzero. Also note that in Magma, “Weierstrass Model” refers to a model with \(a_1=a_2=a_3=0\), which is called Short Weierstrass Model in Sage; these do not always exist in characteristics 2 and 3.
EXAMPLES:
We construct an elliptic curve over an elaborate base ring:
sage: p, a, b = 97, 1, 3
sage: R.<u> = GF(p)[]
sage: S.<v> = R[]
sage: T = S.fraction_field()
sage: E = EllipticCurve(T, [a, b]); E
Elliptic Curve defined by y^2 = x^3 + x + 3 over Fraction Field of Univariate
Polynomial Ring in v over Univariate Polynomial Ring in u over Finite Field of size 97
sage: latex(E)
y^2 = x^{3} + x + 3
>>> from sage.all import *
>>> p, a, b = Integer(97), Integer(1), Integer(3)
>>> R = GF(p)['u']; (u,) = R._first_ngens(1)
>>> S = R['v']; (v,) = S._first_ngens(1)
>>> T = S.fraction_field()
>>> E = EllipticCurve(T, [a, b]); E
Elliptic Curve defined by y^2 = x^3 + x + 3 over Fraction Field of Univariate
Polynomial Ring in v over Univariate Polynomial Ring in u over Finite Field of size 97
>>> latex(E)
y^2 = x^{3} + x + 3
p, a, b = 97, 1, 3 R.<u> = GF(p)[] S.<v> = R[] T = S.fraction_field() E = EllipticCurve(T, [a, b]); E latex(E)
AUTHORS:
William Stein (2005): Initial version
Robert Bradshaw et al….
John Cremona (2008-01): isomorphisms, automorphisms and twists in all characteristics
Julian Rueth (2014-04-11): improved caching
Lorenz Panny (2022-04-14): added
.montgomery_model()
- class sage.schemes.elliptic_curves.ell_generic.EllipticCurve_generic(K, ainvs, category=None)[source]¶
Bases:
WithEqualityById,ProjectivePlaneCurveElliptic curve over a generic base ring.
EXAMPLES:
sage: E = EllipticCurve([1,2,3/4,7,19]); E Elliptic Curve defined by y^2 + x*y + 3/4*y = x^3 + 2*x^2 + 7*x + 19 over Rational Field sage: loads(E.dumps()) == E True sage: E = EllipticCurve([1,3]) sage: P = E([-1,1,1]) sage: -5*P (179051/80089 : -91814227/22665187 : 1)
>>> from sage.all import * >>> E = EllipticCurve([Integer(1),Integer(2),Integer(3)/Integer(4),Integer(7),Integer(19)]); E Elliptic Curve defined by y^2 + x*y + 3/4*y = x^3 + 2*x^2 + 7*x + 19 over Rational Field >>> loads(E.dumps()) == E True >>> E = EllipticCurve([Integer(1),Integer(3)]) >>> P = E([-Integer(1),Integer(1),Integer(1)]) >>> -Integer(5)*P (179051/80089 : -91814227/22665187 : 1)
E = EllipticCurve([1,2,3/4,7,19]); E loads(E.dumps()) == E E = EllipticCurve([1,3]) P = E([-1,1,1]) -5*P
- a1()[source]¶
Return the \(a_1\) invariant of this elliptic curve.
EXAMPLES:
sage: E = EllipticCurve([1,2,3,4,6]) sage: E.a1() 1
>>> from sage.all import * >>> E = EllipticCurve([Integer(1),Integer(2),Integer(3),Integer(4),Integer(6)]) >>> E.a1() 1
E = EllipticCurve([1,2,3,4,6]) E.a1()
- a2()[source]¶
Return the \(a_2\) invariant of this elliptic curve.
EXAMPLES:
sage: E = EllipticCurve([1,2,3,4,6]) sage: E.a2() 2
>>> from sage.all import * >>> E = EllipticCurve([Integer(1),Integer(2),Integer(3),Integer(4),Integer(6)]) >>> E.a2() 2
E = EllipticCurve([1,2,3,4,6]) E.a2()
- a3()[source]¶
Return the \(a_3\) invariant of this elliptic curve.
EXAMPLES:
sage: E = EllipticCurve([1,2,3,4,6]) sage: E.a3() 3
>>> from sage.all import * >>> E = EllipticCurve([Integer(1),Integer(2),Integer(3),Integer(4),Integer(6)]) >>> E.a3() 3
E = EllipticCurve([1,2,3,4,6]) E.a3()
- a4()[source]¶
Return the \(a_4\) invariant of this elliptic curve.
EXAMPLES:
sage: E = EllipticCurve([1,2,3,4,6]) sage: E.a4() 4
>>> from sage.all import * >>> E = EllipticCurve([Integer(1),Integer(2),Integer(3),Integer(4),Integer(6)]) >>> E.a4() 4
E = EllipticCurve([1,2,3,4,6]) E.a4()
- a6()[source]¶
Return the \(a_6\) invariant of this elliptic curve.
EXAMPLES:
sage: E = EllipticCurve([1,2,3,4,6]) sage: E.a6() 6
>>> from sage.all import * >>> E = EllipticCurve([Integer(1),Integer(2),Integer(3),Integer(4),Integer(6)]) >>> E.a6() 6
E = EllipticCurve([1,2,3,4,6]) E.a6()
- a_invariants()[source]¶
The \(a\)-invariants of this elliptic curve, as a tuple.
OUTPUT:
(tuple) - a 5-tuple of the \(a\)-invariants of this elliptic curve.
EXAMPLES:
sage: E = EllipticCurve([1,2,3,4,5]) sage: E.a_invariants() (1, 2, 3, 4, 5) sage: E = EllipticCurve([0,1]); E Elliptic Curve defined by y^2 = x^3 + 1 over Rational Field sage: E.a_invariants() (0, 0, 0, 0, 1) sage: E = EllipticCurve([GF(7)(3),5]) sage: E.a_invariants() (0, 0, 0, 3, 5)
>>> from sage.all import * >>> E = EllipticCurve([Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)]) >>> E.a_invariants() (1, 2, 3, 4, 5) >>> E = EllipticCurve([Integer(0),Integer(1)]); E Elliptic Curve defined by y^2 = x^3 + 1 over Rational Field >>> E.a_invariants() (0, 0, 0, 0, 1) >>> E = EllipticCurve([GF(Integer(7))(Integer(3)),Integer(5)]) >>> E.a_invariants() (0, 0, 0, 3, 5)
E = EllipticCurve([1,2,3,4,5]) E.a_invariants() E = EllipticCurve([0,1]); E E.a_invariants() E = EllipticCurve([GF(7)(3),5]) E.a_invariants()
- ainvs()[source]¶
The \(a\)-invariants of this elliptic curve, as a tuple.
OUTPUT:
(tuple) - a 5-tuple of the \(a\)-invariants of this elliptic curve.
EXAMPLES:
sage: E = EllipticCurve([1,2,3,4,5]) sage: E.a_invariants() (1, 2, 3, 4, 5) sage: E = EllipticCurve([0,1]); E Elliptic Curve defined by y^2 = x^3 + 1 over Rational Field sage: E.a_invariants() (0, 0, 0, 0, 1) sage: E = EllipticCurve([GF(7)(3),5]) sage: E.a_invariants() (0, 0, 0, 3, 5)
>>> from sage.all import * >>> E = EllipticCurve([Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)]) >>> E.a_invariants() (1, 2, 3, 4, 5) >>> E = EllipticCurve([Integer(0),Integer(1)]); E Elliptic Curve defined by y^2 = x^3 + 1 over Rational Field >>> E.a_invariants() (0, 0, 0, 0, 1) >>> E = EllipticCurve([GF(Integer(7))(Integer(3)),Integer(5)]) >>> E.a_invariants() (0, 0, 0, 3, 5)
E = EllipticCurve([1,2,3,4,5]) E.a_invariants() E = EllipticCurve([0,1]); E E.a_invariants() E = EllipticCurve([GF(7)(3),5]) E.a_invariants()
- automorphisms(field=None)[source]¶
Return the set of isomorphisms from
selfto itself (as a list).The identity and negation morphisms are guaranteed to appear as the first and second entry of the returned list.
INPUT:
field– (default:None) a field into which the coefficients of the curve may be coerced (by default, uses the base field of the curve)
OUTPUT:
A list of
WeierstrassIsomorphismobjects consisting of all the isomorphisms from the curveselfto itself defined overfield.EXAMPLES:
sage: E = EllipticCurve_from_j(QQ(0)) # a curve with j=0 over QQ sage: E.automorphisms() [Elliptic-curve endomorphism of Elliptic Curve defined by y^2 + y = x^3 over Rational Field Via: (u,r,s,t) = (1, 0, 0, 0), Elliptic-curve endomorphism of Elliptic Curve defined by y^2 + y = x^3 over Rational Field Via: (u,r,s,t) = (-1, 0, 0, -1)]
>>> from sage.all import * >>> E = EllipticCurve_from_j(QQ(Integer(0))) # a curve with j=0 over QQ >>> E.automorphisms() [Elliptic-curve endomorphism of Elliptic Curve defined by y^2 + y = x^3 over Rational Field Via: (u,r,s,t) = (1, 0, 0, 0), Elliptic-curve endomorphism of Elliptic Curve defined by y^2 + y = x^3 over Rational Field Via: (u,r,s,t) = (-1, 0, 0, -1)]
E = EllipticCurve_from_j(QQ(0)) # a curve with j=0 over QQ E.automorphisms()
We can also find automorphisms defined over extension fields:
sage: x = polygen(ZZ, 'x') sage: K.<a> = NumberField(x^2 + 3) # adjoin roots of unity # needs sage.rings.number_field sage: E.automorphisms(K) # needs sage.rings.number_field [Elliptic-curve endomorphism of Elliptic Curve defined by y^2 + y = x^3 over Number Field in a with defining polynomial x^2 + 3 Via: (u,r,s,t) = (1, 0, 0, 0), Elliptic-curve endomorphism of Elliptic Curve defined by y^2 + y = x^3 over Number Field in a with defining polynomial x^2 + 3 Via: (u,r,s,t) = (-1, 0, 0, -1), Elliptic-curve endomorphism of Elliptic Curve defined by y^2 + y = x^3 over Number Field in a with defining polynomial x^2 + 3 Via: (u,r,s,t) = (-1/2*a - 1/2, 0, 0, 0), Elliptic-curve endomorphism of Elliptic Curve defined by y^2 + y = x^3 over Number Field in a with defining polynomial x^2 + 3 Via: (u,r,s,t) = (1/2*a + 1/2, 0, 0, -1), Elliptic-curve endomorphism of Elliptic Curve defined by y^2 + y = x^3 over Number Field in a with defining polynomial x^2 + 3 Via: (u,r,s,t) = (1/2*a - 1/2, 0, 0, 0), Elliptic-curve endomorphism of Elliptic Curve defined by y^2 + y = x^3 over Number Field in a with defining polynomial x^2 + 3 Via: (u,r,s,t) = (-1/2*a + 1/2, 0, 0, -1)]
>>> from sage.all import * >>> x = polygen(ZZ, 'x') >>> K = NumberField(x**Integer(2) + Integer(3), names=('a',)); (a,) = K._first_ngens(1)# adjoin roots of unity # needs sage.rings.number_field >>> E.automorphisms(K) # needs sage.rings.number_field [Elliptic-curve endomorphism of Elliptic Curve defined by y^2 + y = x^3 over Number Field in a with defining polynomial x^2 + 3 Via: (u,r,s,t) = (1, 0, 0, 0), Elliptic-curve endomorphism of Elliptic Curve defined by y^2 + y = x^3 over Number Field in a with defining polynomial x^2 + 3 Via: (u,r,s,t) = (-1, 0, 0, -1), Elliptic-curve endomorphism of Elliptic Curve defined by y^2 + y = x^3 over Number Field in a with defining polynomial x^2 + 3 Via: (u,r,s,t) = (-1/2*a - 1/2, 0, 0, 0), Elliptic-curve endomorphism of Elliptic Curve defined by y^2 + y = x^3 over Number Field in a with defining polynomial x^2 + 3 Via: (u,r,s,t) = (1/2*a + 1/2, 0, 0, -1), Elliptic-curve endomorphism of Elliptic Curve defined by y^2 + y = x^3 over Number Field in a with defining polynomial x^2 + 3 Via: (u,r,s,t) = (1/2*a - 1/2, 0, 0, 0), Elliptic-curve endomorphism of Elliptic Curve defined by y^2 + y = x^3 over Number Field in a with defining polynomial x^2 + 3 Via: (u,r,s,t) = (-1/2*a + 1/2, 0, 0, -1)]
x = polygen(ZZ, 'x') K.<a> = NumberField(x^2 + 3) # adjoin roots of unity # needs sage.rings.number_field E.automorphisms(K) # needs sage.rings.number_field
sage: [len(EllipticCurve_from_j(GF(q,'a')(0)).automorphisms()) # needs sage.rings.finite_rings ....: for q in [2,4,3,9,5,25,7,49]] [2, 24, 2, 12, 2, 6, 6, 6]
>>> from sage.all import * >>> [len(EllipticCurve_from_j(GF(q,'a')(Integer(0))).automorphisms()) # needs sage.rings.finite_rings ... for q in [Integer(2),Integer(4),Integer(3),Integer(9),Integer(5),Integer(25),Integer(7),Integer(49)]] [2, 24, 2, 12, 2, 6, 6, 6]
[len(EllipticCurve_from_j(GF(q,'a')(0)).automorphisms()) # needs sage.rings.finite_rings for q in [2,4,3,9,5,25,7,49]]
>>> from sage.all import * >>> [len(EllipticCurve_from_j(GF(q,'a')(Integer(0))).automorphisms()) # needs sage.rings.finite_rings ... for q in [Integer(2),Integer(4),Integer(3),Integer(9),Integer(5),Integer(25),Integer(7),Integer(49)]] [2, 24, 2, 12, 2, 6, 6, 6]
[len(EllipticCurve_from_j(GF(q,'a')(0)).automorphisms()) # needs sage.rings.finite_rings for q in [2,4,3,9,5,25,7,49]]
- b2()[source]¶
Return the \(b_2\) invariant of this elliptic curve.
EXAMPLES:
sage: E = EllipticCurve([1,2,3,4,5]) sage: E.b2() 9
>>> from sage.all import * >>> E = EllipticCurve([Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)]) >>> E.b2() 9
E = EllipticCurve([1,2,3,4,5]) E.b2()
- b4()[source]¶
Return the \(b_4\) invariant of this elliptic curve.
EXAMPLES:
sage: E = EllipticCurve([1,2,3,4,5]) sage: E.b4() 11
>>> from sage.all import * >>> E = EllipticCurve([Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)]) >>> E.b4() 11
E = EllipticCurve([1,2,3,4,5]) E.b4()
- b6()[source]¶
Return the \(b_6\) invariant of this elliptic curve.
EXAMPLES:
sage: E = EllipticCurve([1,2,3,4,5]) sage: E.b6() 29
>>> from sage.all import * >>> E = EllipticCurve([Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)]) >>> E.b6() 29
E = EllipticCurve([1,2,3,4,5]) E.b6()
- b8()[source]¶
Return the \(b_8\) invariant of this elliptic curve.
EXAMPLES:
sage: E = EllipticCurve([1,2,3,4,5]) sage: E.b8() 35
>>> from sage.all import * >>> E = EllipticCurve([Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)]) >>> E.b8() 35
E = EllipticCurve([1,2,3,4,5]) E.b8()
- b_invariants()[source]¶
Return the \(b\)-invariants of this elliptic curve, as a tuple.
OUTPUT:
(tuple) - a 4-tuple of the \(b\)-invariants of this elliptic curve.
This method is cached.
EXAMPLES:
sage: E = EllipticCurve([0, -1, 1, -10, -20]) sage: E.b_invariants() (-4, -20, -79, -21) sage: E = EllipticCurve([-4,0]) sage: E.b_invariants() (0, -8, 0, -16) sage: E = EllipticCurve([1,2,3,4,5]) sage: E.b_invariants() (9, 11, 29, 35) sage: E.b2() 9 sage: E.b4() 11 sage: E.b6() 29 sage: E.b8() 35
>>> from sage.all import * >>> E = EllipticCurve([Integer(0), -Integer(1), Integer(1), -Integer(10), -Integer(20)]) >>> E.b_invariants() (-4, -20, -79, -21) >>> E = EllipticCurve([-Integer(4),Integer(0)]) >>> E.b_invariants() (0, -8, 0, -16) >>> E = EllipticCurve([Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)]) >>> E.b_invariants() (9, 11, 29, 35) >>> E.b2() 9 >>> E.b4() 11 >>> E.b6() 29 >>> E.b8() 35
E = EllipticCurve([0, -1, 1, -10, -20]) E.b_invariants() E = EllipticCurve([-4,0]) E.b_invariants() E = EllipticCurve([1,2,3,4,5]) E.b_invariants() E.b2() E.b4() E.b6() E.b8()
ALGORITHM:
These are simple functions of the \(a\)-invariants.
AUTHORS:
William Stein (2005-04-25)
- base_extend(R)[source]¶
Return the base extension of
selfto \(R\).INPUT:
R– either a ring into which the \(a\)-invariants ofselfmay be converted, or a morphism which may be applied to them.
OUTPUT:
An elliptic curve over the new ring whose \(a\)-invariants are the images of the \(a\)-invariants of
self.EXAMPLES:
sage: E = EllipticCurve(GF(5), [1,1]); E Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field of size 5 sage: E1 = E.base_extend(GF(125,'a')); E1 # needs sage.rings.finite_rings Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field in a of size 5^3
>>> from sage.all import * >>> E = EllipticCurve(GF(Integer(5)), [Integer(1),Integer(1)]); E Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field of size 5 >>> E1 = E.base_extend(GF(Integer(125),'a')); E1 # needs sage.rings.finite_rings Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field in a of size 5^3
E = EllipticCurve(GF(5), [1,1]); E E1 = E.base_extend(GF(125,'a')); E1 # needs sage.rings.finite_rings
- base_ring()[source]¶
Return the base ring of the elliptic curve.
EXAMPLES:
sage: E = EllipticCurve(GF(49, 'a'), [3,5]) # needs sage.rings.finite_rings sage: E.base_ring() # needs sage.rings.finite_rings Finite Field in a of size 7^2
>>> from sage.all import * >>> E = EllipticCurve(GF(Integer(49), 'a'), [Integer(3),Integer(5)]) # needs sage.rings.finite_rings >>> E.base_ring() # needs sage.rings.finite_rings Finite Field in a of size 7^2
E = EllipticCurve(GF(49, 'a'), [3,5]) # needs sage.rings.finite_rings E.base_ring() # needs sage.rings.finite_rings
sage: E = EllipticCurve([1,1]) sage: E.base_ring() Rational Field
>>> from sage.all import * >>> E = EllipticCurve([Integer(1),Integer(1)]) >>> E.base_ring() Rational Field
E = EllipticCurve([1,1]) E.base_ring()
>>> from sage.all import * >>> E = EllipticCurve([Integer(1),Integer(1)]) >>> E.base_ring() Rational Field
E = EllipticCurve([1,1]) E.base_ring()
sage: E = EllipticCurve(ZZ, [3,5]) sage: E.base_ring() Integer Ring
>>> from sage.all import * >>> E = EllipticCurve(ZZ, [Integer(3),Integer(5)]) >>> E.base_ring() Integer Ring
E = EllipticCurve(ZZ, [3,5]) E.base_ring()
>>> from sage.all import * >>> E = EllipticCurve(ZZ, [Integer(3),Integer(5)]) >>> E.base_ring() Integer Ring
E = EllipticCurve(ZZ, [3,5]) E.base_ring()
- c4()[source]¶
Return the \(c_4\) invariant of this elliptic curve.
EXAMPLES:
sage: E = EllipticCurve([0, -1, 1, -10, -20]) sage: E.c4() 496
>>> from sage.all import * >>> E = EllipticCurve([Integer(0), -Integer(1), Integer(1), -Integer(10), -Integer(20)]) >>> E.c4() 496
E = EllipticCurve([0, -1, 1, -10, -20]) E.c4()
- c6()[source]¶
Return the \(c_6\) invariant of this elliptic curve.
EXAMPLES:
sage: E = EllipticCurve([0, -1, 1, -10, -20]) sage: E.c6() 20008
>>> from sage.all import * >>> E = EllipticCurve([Integer(0), -Integer(1), Integer(1), -Integer(10), -Integer(20)]) >>> E.c6() 20008
E = EllipticCurve([0, -1, 1, -10, -20]) E.c6()
- c_invariants()[source]¶
Return the \(c\)-invariants of this elliptic curve, as a tuple.
This method is cached.
OUTPUT:
(tuple) - a 2-tuple of the \(c\)-invariants of the elliptic curve.
EXAMPLES:
sage: E = EllipticCurve([0, -1, 1, -10, -20]) sage: E.c_invariants() (496, 20008) sage: E = EllipticCurve([-4,0]) sage: E.c_invariants() (192, 0)
>>> from sage.all import * >>> E = EllipticCurve([Integer(0), -Integer(1), Integer(1), -Integer(10), -Integer(20)]) >>> E.c_invariants() (496, 20008) >>> E = EllipticCurve([-Integer(4),Integer(0)]) >>> E.c_invariants() (192, 0)
E = EllipticCurve([0, -1, 1, -10, -20]) E.c_invariants() E = EllipticCurve([-4,0]) E.c_invariants()
ALGORITHM:
These are simple functions of the \(a\)-invariants.
AUTHORS:
William Stein (2005-04-25)
- change_ring(R)[source]¶
Return the base change of
selfto \(R\).This has the same effect as
self.base_extend(R).EXAMPLES:
sage: # needs sage.rings.finite_rings sage: F2 = GF(5^2,'a'); a = F2.gen() sage: F4 = GF(5^4,'b'); b = F4.gen() sage: roots = a.charpoly().roots(ring=F4, multiplicities=False) sage: h = F2.hom([roots[0]], F4) sage: E = EllipticCurve(F2, [1,a]); E Elliptic Curve defined by y^2 = x^3 + x + a over Finite Field in a of size 5^2 sage: E.change_ring(h) Elliptic Curve defined by y^2 = x^3 + x + (4*b^3+4*b^2+4*b+3) over Finite Field in b of size 5^4
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> F2 = GF(Integer(5)**Integer(2),'a'); a = F2.gen() >>> F4 = GF(Integer(5)**Integer(4),'b'); b = F4.gen() >>> roots = a.charpoly().roots(ring=F4, multiplicities=False) >>> h = F2.hom([roots[Integer(0)]], F4) >>> E = EllipticCurve(F2, [Integer(1),a]); E Elliptic Curve defined by y^2 = x^3 + x + a over Finite Field in a of size 5^2 >>> E.change_ring(h) Elliptic Curve defined by y^2 = x^3 + x + (4*b^3+4*b^2+4*b+3) over Finite Field in b of size 5^4
# needs sage.rings.finite_rings F2 = GF(5^2,'a'); a = F2.gen() F4 = GF(5^4,'b'); b = F4.gen() roots = a.charpoly().roots(ring=F4, multiplicities=False) h = F2.hom([roots[0]], F4) E = EllipticCurve(F2, [1,a]); E E.change_ring(h)
- change_weierstrass_model(*urst)[source]¶
Return a new Weierstrass model of
selfunder the standard transformation \((u,r,s,t)\).\[(x,y) \mapsto (x',y') = (u^2x + r , u^3y + su^2x + t).\]EXAMPLES:
sage: E = EllipticCurve('15a') sage: F1 = E.change_weierstrass_model([1/2,0,0,0]); F1 Elliptic Curve defined by y^2 + 2*x*y + 8*y = x^3 + 4*x^2 - 160*x - 640 over Rational Field sage: F2 = E.change_weierstrass_model([7,2,1/3,5]); F2 Elliptic Curve defined by y^2 + 5/21*x*y + 13/343*y = x^3 + 59/441*x^2 - 10/7203*x - 58/117649 over Rational Field sage: F1.is_isomorphic(F2) True
>>> from sage.all import * >>> E = EllipticCurve('15a') >>> F1 = E.change_weierstrass_model([Integer(1)/Integer(2),Integer(0),Integer(0),Integer(0)]); F1 Elliptic Curve defined by y^2 + 2*x*y + 8*y = x^3 + 4*x^2 - 160*x - 640 over Rational Field >>> F2 = E.change_weierstrass_model([Integer(7),Integer(2),Integer(1)/Integer(3),Integer(5)]); F2 Elliptic Curve defined by y^2 + 5/21*x*y + 13/343*y = x^3 + 59/441*x^2 - 10/7203*x - 58/117649 over Rational Field >>> F1.is_isomorphic(F2) True
E = EllipticCurve('15a') F1 = E.change_weierstrass_model([1/2,0,0,0]); F1 F2 = E.change_weierstrass_model([7,2,1/3,5]); F2 F1.is_isomorphic(F2)
- discriminant()[source]¶
Return the discriminant of this elliptic curve.
This method is cached.
EXAMPLES:
sage: E = EllipticCurve([0,0,1,-1,0]) sage: E.discriminant() 37 sage: E = EllipticCurve([0, -1, 1, -10, -20]) sage: E.discriminant() -161051 sage: E = EllipticCurve([GF(7)(2),1]) sage: E.discriminant() 1
>>> from sage.all import * >>> E = EllipticCurve([Integer(0),Integer(0),Integer(1),-Integer(1),Integer(0)]) >>> E.discriminant() 37 >>> E = EllipticCurve([Integer(0), -Integer(1), Integer(1), -Integer(10), -Integer(20)]) >>> E.discriminant() -161051 >>> E = EllipticCurve([GF(Integer(7))(Integer(2)),Integer(1)]) >>> E.discriminant() 1
E = EllipticCurve([0,0,1,-1,0]) E.discriminant() E = EllipticCurve([0, -1, 1, -10, -20]) E.discriminant() E = EllipticCurve([GF(7)(2),1]) E.discriminant()
- division_polynomial(m, x=None, two_torsion_multiplicity=2, force_evaluate=None)[source]¶
Return the \(m\)-th division polynomial of this elliptic curve evaluated at \(x\).
The division polynomial is cached if \(x\) is
None.INPUT:
m– positive integerx– (optional) ring element to use as the \(x\) variable. If \(x\) isNone(omitted), then a new polynomial ring will be constructed over the base ring of the elliptic curve, and its generator will be used as \(x\). Note that \(x\) does not need to be a generator of a polynomial ring; any ring element works. This permits fast calculation of the torsion polynomial evaluated on any element of a ring.two_torsion_multiplicity– 0, 1, or 2If 0: For even \(m\) when \(x\) is
None, a univariate polynomial over the base ring of the curve is returned, which omits factors whose roots are the \(x\)-coordinates of the \(2\)-torsion points. When \(x\) is notNone, the evaluation of such a polynomial at \(x\) is returned.If 2: For even \(m\) when \(x\) is
None, a univariate polynomial over the base ring of the curve is returned, which includes a factor of degree 3 whose roots are the \(x\)-coordinates of the \(2\)-torsion points. Similarly, when \(x\) is notNone, the evaluation of such a polynomial at \(x\) is returned.If 1: For even \(m\) when \(x\) is
None, a bivariate polynomial over the base ring of the curve is returned, which includes a factor \(2y+a_1x+a_3\) having simple zeros at the \(2\)-torsion points. When \(x\) is notNone, it should be a tuple of length 2, and the evaluation of such a polynomial at \(x\) is returned.force_evaluate– (optional) 0, 1, or 2By default, this method makes use of previously cached generic division polynomials to compute the value of the polynomial at a given element \(x\) whenever it appears beneficial to do so. Explicitly setting this flag overrides the default behavior.
Note that the complexity of evaluating a generic division polynomial scales much worse than that of computing the value at a point directly (using the recursive formulas), hence setting this flag can be detrimental to performance.
If 0: Do not use cached generic division polynomials.
If 1: If the generic division polynomial for this \(m\) has been cached before, evaluate it at \(x\) to compute the result.
If 2: Compute the value at \(x\) by evaluating the generic division polynomial. If the generic \(m\)-division polynomial has not yet been cached, compute and cache it first.
EXAMPLES:
sage: E = EllipticCurve([0,0,1,-1,0]) sage: E.division_polynomial(1) 1 sage: E.division_polynomial(2, two_torsion_multiplicity=0) 1 sage: E.division_polynomial(2, two_torsion_multiplicity=1) 2*y + 1 sage: E.division_polynomial(2, two_torsion_multiplicity=2) 4*x^3 - 4*x + 1 sage: E.division_polynomial(2) 4*x^3 - 4*x + 1 sage: [E.division_polynomial(3, two_torsion_multiplicity=i) for i in range(3)] [3*x^4 - 6*x^2 + 3*x - 1, 3*x^4 - 6*x^2 + 3*x - 1, 3*x^4 - 6*x^2 + 3*x - 1] sage: [type(E.division_polynomial(3, two_torsion_multiplicity=i)) for i in range(3)] [<... 'sage.rings.polynomial.polynomial_rational_flint.Polynomial_rational_flint'>, <... 'sage.rings.polynomial.multi_polynomial_libsingular.MPolynomial_libsingular'>, <... 'sage.rings.polynomial.polynomial_rational_flint.Polynomial_rational_flint'>]
>>> from sage.all import * >>> E = EllipticCurve([Integer(0),Integer(0),Integer(1),-Integer(1),Integer(0)]) >>> E.division_polynomial(Integer(1)) 1 >>> E.division_polynomial(Integer(2), two_torsion_multiplicity=Integer(0)) 1 >>> E.division_polynomial(Integer(2), two_torsion_multiplicity=Integer(1)) 2*y + 1 >>> E.division_polynomial(Integer(2), two_torsion_multiplicity=Integer(2)) 4*x^3 - 4*x + 1 >>> E.division_polynomial(Integer(2)) 4*x^3 - 4*x + 1 >>> [E.division_polynomial(Integer(3), two_torsion_multiplicity=i) for i in range(Integer(3))] [3*x^4 - 6*x^2 + 3*x - 1, 3*x^4 - 6*x^2 + 3*x - 1, 3*x^4 - 6*x^2 + 3*x - 1] >>> [type(E.division_polynomial(Integer(3), two_torsion_multiplicity=i)) for i in range(Integer(3))] [<... 'sage.rings.polynomial.polynomial_rational_flint.Polynomial_rational_flint'>, <... 'sage.rings.polynomial.multi_polynomial_libsingular.MPolynomial_libsingular'>, <... 'sage.rings.polynomial.polynomial_rational_flint.Polynomial_rational_flint'>]
E = EllipticCurve([0,0,1,-1,0]) E.division_polynomial(1) E.division_polynomial(2, two_torsion_multiplicity=0) E.division_polynomial(2, two_torsion_multiplicity=1) E.division_polynomial(2, two_torsion_multiplicity=2) E.division_polynomial(2) [E.division_polynomial(3, two_torsion_multiplicity=i) for i in range(3)] [type(E.division_polynomial(3, two_torsion_multiplicity=i)) for i in range(3)]
sage: E = EllipticCurve([0, -1, 1, -10, -20]) sage: R.<z> = PolynomialRing(QQ) sage: E.division_polynomial(4, z, 0) 2*z^6 - 4*z^5 - 100*z^4 - 790*z^3 - 210*z^2 - 1496*z - 5821 sage: E.division_polynomial(4, z) 8*z^9 - 24*z^8 - 464*z^7 - 2758*z^6 + 6636*z^5 + 34356*z^4 + 53510*z^3 + 99714*z^2 + 351024*z + 459859
>>> from sage.all import * >>> E = EllipticCurve([Integer(0), -Integer(1), Integer(1), -Integer(10), -Integer(20)]) >>> R = PolynomialRing(QQ, names=('z',)); (z,) = R._first_ngens(1) >>> E.division_polynomial(Integer(4), z, Integer(0)) 2*z^6 - 4*z^5 - 100*z^4 - 790*z^3 - 210*z^2 - 1496*z - 5821 >>> E.division_polynomial(Integer(4), z) 8*z^9 - 24*z^8 - 464*z^7 - 2758*z^6 + 6636*z^5 + 34356*z^4 + 53510*z^3 + 99714*z^2 + 351024*z + 459859
E = EllipticCurve([0, -1, 1, -10, -20]) R.<z> = PolynomialRing(QQ) E.division_polynomial(4, z, 0) E.division_polynomial(4, z)
>>> from sage.all import * >>> E = EllipticCurve([Integer(0), -Integer(1), Integer(1), -Integer(10), -Integer(20)]) >>> R = PolynomialRing(QQ, names=('z',)); (z,) = R._first_ngens(1) >>> E.division_polynomial(Integer(4), z, Integer(0)) 2*z^6 - 4*z^5 - 100*z^4 - 790*z^3 - 210*z^2 - 1496*z - 5821 >>> E.division_polynomial(Integer(4), z) 8*z^9 - 24*z^8 - 464*z^7 - 2758*z^6 + 6636*z^5 + 34356*z^4 + 53510*z^3 + 99714*z^2 + 351024*z + 459859
E = EllipticCurve([0, -1, 1, -10, -20]) R.<z> = PolynomialRing(QQ) E.division_polynomial(4, z, 0) E.division_polynomial(4, z)
This does not work, since when two_torsion_multiplicity is 1, we compute a bivariate polynomial, and must evaluate at a tuple of length 2:
sage: E.division_polynomial(4,z,1) Traceback (most recent call last): ... ValueError: x should be a tuple of length 2 (or None) when two_torsion_multiplicity is 1 sage: R.<z,w> = PolynomialRing(QQ, 2) sage: E.division_polynomial(4, (z,w), 1).factor() (2*w + 1) * (2*z^6 - 4*z^5 - 100*z^4 - 790*z^3 - 210*z^2 - 1496*z - 5821)
>>> from sage.all import * >>> E.division_polynomial(Integer(4),z,Integer(1)) Traceback (most recent call last): ... ValueError: x should be a tuple of length 2 (or None) when two_torsion_multiplicity is 1 >>> R = PolynomialRing(QQ, Integer(2), names=('z', 'w',)); (z, w,) = R._first_ngens(2) >>> E.division_polynomial(Integer(4), (z,w), Integer(1)).factor() (2*w + 1) * (2*z^6 - 4*z^5 - 100*z^4 - 790*z^3 - 210*z^2 - 1496*z - 5821)
E.division_polynomial(4,z,1) R.<z,w> = PolynomialRing(QQ, 2) E.division_polynomial(4, (z,w), 1).factor()
We can also evaluate this bivariate polynomial at a point:
sage: P = E(5,5) sage: E.division_polynomial(4,P,two_torsion_multiplicity=1) -1771561
>>> from sage.all import * >>> P = E(Integer(5),Integer(5)) >>> E.division_polynomial(Integer(4),P,two_torsion_multiplicity=Integer(1)) -1771561
P = E(5,5) E.division_polynomial(4,P,two_torsion_multiplicity=1)
- division_polynomial_0(n, x=None)[source]¶
Return the \(n\)-th torsion (division) polynomial, without the 2-torsion factor if \(n\) is even, as a polynomial in \(x\).
These are the polynomials \(g_n\) defined in [MT1991], but with the sign flipped for even \(n\), so that the leading coefficient is always positive.
Note
This function is intended for internal use; users should use
division_polynomial().See also
_multiple_x_numerator()_multiple_x_denominator()
INPUT:
n– positive integer, or the special values-1and-2which mean \(B_6 = (2y + a_1 x + a_3)^2\) and \(B_6^2\) respectively (in the notation of [MT1991]), or a list of integersx– a ring element to use as the “x” variable orNone(default:None); ifNone, then a new polynomial ring will be constructed over the base ring of the elliptic curve, and its generator will be used asx. Note thatxdoes not need to be a generator of a polynomial ring; any ring element is ok. This permits fast calculation of the torsion polynomial evaluated on any element of a ring.
ALGORITHM:
Recursion described in [MT1991]. The recursive formulae are evaluated \(O(\log^2 n)\) times.
AUTHORS:
David Harvey (2006-09-24): initial version
John Cremona (2008-08-26): unified division polynomial code
EXAMPLES:
sage: E = EllipticCurve("37a") sage: E.division_polynomial_0(1) 1 sage: E.division_polynomial_0(2) 1 sage: E.division_polynomial_0(3) 3*x^4 - 6*x^2 + 3*x - 1 sage: E.division_polynomial_0(4) 2*x^6 - 10*x^4 + 10*x^3 - 10*x^2 + 2*x + 1 sage: E.division_polynomial_0(5) 5*x^12 - 62*x^10 + 95*x^9 - 105*x^8 - 60*x^7 + 285*x^6 - 174*x^5 - 5*x^4 - 5*x^3 + 35*x^2 - 15*x + 2 sage: E.division_polynomial_0(6) 3*x^16 - 72*x^14 + 168*x^13 - 364*x^12 + 1120*x^10 - 1144*x^9 + 300*x^8 - 540*x^7 + 1120*x^6 - 588*x^5 - 133*x^4 + 252*x^3 - 114*x^2 + 22*x - 1 sage: E.division_polynomial_0(7) 7*x^24 - 308*x^22 + 986*x^21 - 2954*x^20 + 28*x^19 + 17171*x^18 - 23142*x^17 + 511*x^16 - 5012*x^15 + 43804*x^14 - 7140*x^13 - 96950*x^12 + 111356*x^11 - 19516*x^10 - 49707*x^9 + 40054*x^8 - 124*x^7 - 18382*x^6 + 13342*x^5 - 4816*x^4 + 1099*x^3 - 210*x^2 + 35*x - 3 sage: E.division_polynomial_0(8) 4*x^30 - 292*x^28 + 1252*x^27 - 5436*x^26 + 2340*x^25 + 39834*x^24 - 79560*x^23 + 51432*x^22 - 142896*x^21 + 451596*x^20 - 212040*x^19 - 1005316*x^18 + 1726416*x^17 - 671160*x^16 - 954924*x^15 + 1119552*x^14 + 313308*x^13 - 1502818*x^12 + 1189908*x^11 - 160152*x^10 - 399176*x^9 + 386142*x^8 - 220128*x^7 + 99558*x^6 - 33528*x^5 + 6042*x^4 + 310*x^3 - 406*x^2 + 78*x - 5
>>> from sage.all import * >>> E = EllipticCurve("37a") >>> E.division_polynomial_0(Integer(1)) 1 >>> E.division_polynomial_0(Integer(2)) 1 >>> E.division_polynomial_0(Integer(3)) 3*x^4 - 6*x^2 + 3*x - 1 >>> E.division_polynomial_0(Integer(4)) 2*x^6 - 10*x^4 + 10*x^3 - 10*x^2 + 2*x + 1 >>> E.division_polynomial_0(Integer(5)) 5*x^12 - 62*x^10 + 95*x^9 - 105*x^8 - 60*x^7 + 285*x^6 - 174*x^5 - 5*x^4 - 5*x^3 + 35*x^2 - 15*x + 2 >>> E.division_polynomial_0(Integer(6)) 3*x^16 - 72*x^14 + 168*x^13 - 364*x^12 + 1120*x^10 - 1144*x^9 + 300*x^8 - 540*x^7 + 1120*x^6 - 588*x^5 - 133*x^4 + 252*x^3 - 114*x^2 + 22*x - 1 >>> E.division_polynomial_0(Integer(7)) 7*x^24 - 308*x^22 + 986*x^21 - 2954*x^20 + 28*x^19 + 17171*x^18 - 23142*x^17 + 511*x^16 - 5012*x^15 + 43804*x^14 - 7140*x^13 - 96950*x^12 + 111356*x^11 - 19516*x^10 - 49707*x^9 + 40054*x^8 - 124*x^7 - 18382*x^6 + 13342*x^5 - 4816*x^4 + 1099*x^3 - 210*x^2 + 35*x - 3 >>> E.division_polynomial_0(Integer(8)) 4*x^30 - 292*x^28 + 1252*x^27 - 5436*x^26 + 2340*x^25 + 39834*x^24 - 79560*x^23 + 51432*x^22 - 142896*x^21 + 451596*x^20 - 212040*x^19 - 1005316*x^18 + 1726416*x^17 - 671160*x^16 - 954924*x^15 + 1119552*x^14 + 313308*x^13 - 1502818*x^12 + 1189908*x^11 - 160152*x^10 - 399176*x^9 + 386142*x^8 - 220128*x^7 + 99558*x^6 - 33528*x^5 + 6042*x^4 + 310*x^3 - 406*x^2 + 78*x - 5
E = EllipticCurve("37a") E.division_polynomial_0(1) E.division_polynomial_0(2) E.division_polynomial_0(3) E.division_polynomial_0(4) E.division_polynomial_0(5) E.division_polynomial_0(6) E.division_polynomial_0(7) E.division_polynomial_0(8)sage: E.division_polynomial_0(18) % E.division_polynomial_0(6) == 0 True
>>> from sage.all import * >>> E.division_polynomial_0(Integer(18)) % E.division_polynomial_0(Integer(6)) == Integer(0) True
E.division_polynomial_0(18) % E.division_polynomial_0(6) == 0
>>> from sage.all import * >>> E.division_polynomial_0(Integer(18)) % E.division_polynomial_0(Integer(6)) == Integer(0) True
E.division_polynomial_0(18) % E.division_polynomial_0(6) == 0
An example to illustrate the relationship with torsion points:
sage: F = GF(11) sage: E = EllipticCurve(F, [0, 2]); E Elliptic Curve defined by y^2 = x^3 + 2 over Finite Field of size 11 sage: f = E.division_polynomial_0(5); f 5*x^12 + x^9 + 8*x^6 + 4*x^3 + 7 sage: f.factor() (5) * (x^2 + 5) * (x^2 + 2*x + 5) * (x^2 + 5*x + 7) * (x^2 + 7*x + 7) * (x^2 + 9*x + 5) * (x^2 + 10*x + 7)
>>> from sage.all import * >>> F = GF(Integer(11)) >>> E = EllipticCurve(F, [Integer(0), Integer(2)]); E Elliptic Curve defined by y^2 = x^3 + 2 over Finite Field of size 11 >>> f = E.division_polynomial_0(Integer(5)); f 5*x^12 + x^9 + 8*x^6 + 4*x^3 + 7 >>> f.factor() (5) * (x^2 + 5) * (x^2 + 2*x + 5) * (x^2 + 5*x + 7) * (x^2 + 7*x + 7) * (x^2 + 9*x + 5) * (x^2 + 10*x + 7)
F = GF(11) E = EllipticCurve(F, [0, 2]); E f = E.division_polynomial_0(5); f f.factor()
This indicates that the \(x\)-coordinates of all the 5-torsion points of \(E\) are in \(\GF{11^2}\), and therefore the \(y\)-coordinates are in \(\GF{11^4}\):
sage: # needs sage.rings.finite_rings sage: K = GF(11^4, 'a') sage: X = E.change_ring(K) sage: f = X.division_polynomial_0(5) sage: x_coords = f.roots(multiplicities=False); x_coords [10*a^3 + 4*a^2 + 5*a + 6, 9*a^3 + 8*a^2 + 10*a + 8, 8*a^3 + a^2 + 4*a + 10, 8*a^3 + a^2 + 4*a + 8, 8*a^3 + a^2 + 4*a + 4, 6*a^3 + 9*a^2 + 3*a + 4, 5*a^3 + 2*a^2 + 8*a + 7, 3*a^3 + 10*a^2 + 7*a + 8, 3*a^3 + 10*a^2 + 7*a + 3, 3*a^3 + 10*a^2 + 7*a + 1, 2*a^3 + 3*a^2 + a + 7, a^3 + 7*a^2 + 6*a]
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> K = GF(Integer(11)**Integer(4), 'a') >>> X = E.change_ring(K) >>> f = X.division_polynomial_0(Integer(5)) >>> x_coords = f.roots(multiplicities=False); x_coords [10*a^3 + 4*a^2 + 5*a + 6, 9*a^3 + 8*a^2 + 10*a + 8, 8*a^3 + a^2 + 4*a + 10, 8*a^3 + a^2 + 4*a + 8, 8*a^3 + a^2 + 4*a + 4, 6*a^3 + 9*a^2 + 3*a + 4, 5*a^3 + 2*a^2 + 8*a + 7, 3*a^3 + 10*a^2 + 7*a + 8, 3*a^3 + 10*a^2 + 7*a + 3, 3*a^3 + 10*a^2 + 7*a + 1, 2*a^3 + 3*a^2 + a + 7, a^3 + 7*a^2 + 6*a]
# needs sage.rings.finite_rings K = GF(11^4, 'a') X = E.change_ring(K) f = X.division_polynomial_0(5) x_coords = f.roots(multiplicities=False); x_coords
Now we check that these are exactly the \(x\)-coordinates of the 5-torsion points of \(E\):
sage: for x in x_coords: # needs sage.rings.finite_rings ....: assert X.lift_x(x).order() == 5
>>> from sage.all import * >>> for x in x_coords: # needs sage.rings.finite_rings ... assert X.lift_x(x).order() == Integer(5)
for x in x_coords: # needs sage.rings.finite_rings assert X.lift_x(x).order() == 5The roots of the polynomial are the \(x\)-coordinates of the points \(P\) such that \(mP=0\) but \(2P\not=0\):
sage: E = EllipticCurve('14a1') sage: T = E.torsion_subgroup() sage: [n*T.0 for n in range(6)] [(0 : 1 : 0), (9 : 23 : 1), (2 : 2 : 1), (1 : -1 : 1), (2 : -5 : 1), (9 : -33 : 1)] sage: pol = E.division_polynomial_0(6) sage: xlist = pol.roots(multiplicities=False); xlist [9, 2, -1/3, -5] sage: [E.lift_x(x, all=True) for x in xlist] [[(9 : -33 : 1), (9 : 23 : 1)], [(2 : -5 : 1), (2 : 2 : 1)], [], []]
>>> from sage.all import * >>> E = EllipticCurve('14a1') >>> T = E.torsion_subgroup() >>> [n*T.gen(0) for n in range(Integer(6))] [(0 : 1 : 0), (9 : 23 : 1), (2 : 2 : 1), (1 : -1 : 1), (2 : -5 : 1), (9 : -33 : 1)] >>> pol = E.division_polynomial_0(Integer(6)) >>> xlist = pol.roots(multiplicities=False); xlist [9, 2, -1/3, -5] >>> [E.lift_x(x, all=True) for x in xlist] [[(9 : -33 : 1), (9 : 23 : 1)], [(2 : -5 : 1), (2 : 2 : 1)], [], []]
E = EllipticCurve('14a1') T = E.torsion_subgroup() [n*T.0 for n in range(6)] pol = E.division_polynomial_0(6) xlist = pol.roots(multiplicities=False); xlist [E.lift_x(x, all=True) for x in xlist]Note
The point of order 2 and the identity do not appear. The points with \(x=-1/3\) and \(x=-5\) are not rational.
- formal()[source]¶
Return the formal group associated to this elliptic curve.
This method is cached.
EXAMPLES:
sage: E = EllipticCurve("37a") sage: E.formal_group() Formal Group associated to the Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
>>> from sage.all import * >>> E = EllipticCurve("37a") >>> E.formal_group() Formal Group associated to the Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
E = EllipticCurve("37a") E.formal_group()
- formal_group()[source]¶
Return the formal group associated to this elliptic curve.
This method is cached.
EXAMPLES:
sage: E = EllipticCurve("37a") sage: E.formal_group() Formal Group associated to the Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
>>> from sage.all import * >>> E = EllipticCurve("37a") >>> E.formal_group() Formal Group associated to the Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
E = EllipticCurve("37a") E.formal_group()
- frobenius_isogeny(n=1)[source]¶
Return the \(n\)-power Frobenius isogeny from this curve to its Galois conjugate.
The Frobenius endomorphism is the special case where \(n\) is divisible by the degree of the base ring of the curve.
See also
EXAMPLES:
sage: # needs sage.rings.finite_rings sage: z3, = GF(13^3).gens() sage: E = EllipticCurve([z3, z3^2]) sage: E.frobenius_isogeny() Frobenius isogeny of degree 13: From: Elliptic Curve defined by y^2 = x^3 + z3*x + z3^2 over Finite Field in z3 of size 13^3 To: Elliptic Curve defined by y^2 = x^3 + (5*z3^2+7*z3+11)*x + (5*z3^2+12*z3+1) over Finite Field in z3 of size 13^3 sage: E.frobenius_isogeny(3) Frobenius endomorphism of degree 2197 = 13^3: From: Elliptic Curve defined by y^2 = x^3 + z3*x + z3^2 over Finite Field in z3 of size 13^3 To: Elliptic Curve defined by y^2 = x^3 + z3*x + z3^2 over Finite Field in z3 of size 13^3
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> z3, = GF(Integer(13)**Integer(3)).gens() >>> E = EllipticCurve([z3, z3**Integer(2)]) >>> E.frobenius_isogeny() Frobenius isogeny of degree 13: From: Elliptic Curve defined by y^2 = x^3 + z3*x + z3^2 over Finite Field in z3 of size 13^3 To: Elliptic Curve defined by y^2 = x^3 + (5*z3^2+7*z3+11)*x + (5*z3^2+12*z3+1) over Finite Field in z3 of size 13^3 >>> E.frobenius_isogeny(Integer(3)) Frobenius endomorphism of degree 2197 = 13^3: From: Elliptic Curve defined by y^2 = x^3 + z3*x + z3^2 over Finite Field in z3 of size 13^3 To: Elliptic Curve defined by y^2 = x^3 + z3*x + z3^2 over Finite Field in z3 of size 13^3
# needs sage.rings.finite_rings z3, = GF(13^3).gens() E = EllipticCurve([z3, z3^2]) E.frobenius_isogeny() E.frobenius_isogeny(3)
- gen(i)[source]¶
Function returning the i’th generator of this elliptic curve.
Note
Relies on gens() being implemented.
EXAMPLES:
sage: R.<a1,a2,a3,a4,a6> = QQ[] sage: E = EllipticCurve([a1,a2,a3,a4,a6]) sage: E.gen(0) Traceback (most recent call last): ... NotImplementedError: not implemented.
>>> from sage.all import * >>> R = QQ['a1, a2, a3, a4, a6']; (a1, a2, a3, a4, a6,) = R._first_ngens(5) >>> E = EllipticCurve([a1,a2,a3,a4,a6]) >>> E.gen(Integer(0)) Traceback (most recent call last): ... NotImplementedError: not implemented.
R.<a1,a2,a3,a4,a6> = QQ[] E = EllipticCurve([a1,a2,a3,a4,a6]) E.gen(0)
- gens()[source]¶
Placeholder function to return generators of an elliptic curve.
Note
This functionality is implemented in certain derived classes, such as EllipticCurve_rational_field.
EXAMPLES:
sage: R.<a1,a2,a3,a4,a6> = QQ[] sage: E = EllipticCurve([a1,a2,a3,a4,a6]) sage: E.gens() Traceback (most recent call last): ... NotImplementedError: not implemented. sage: E = EllipticCurve(QQ, [1,1]) sage: E.gens() [(0 : 1 : 1)]
>>> from sage.all import * >>> R = QQ['a1, a2, a3, a4, a6']; (a1, a2, a3, a4, a6,) = R._first_ngens(5) >>> E = EllipticCurve([a1,a2,a3,a4,a6]) >>> E.gens() Traceback (most recent call last): ... NotImplementedError: not implemented. >>> E = EllipticCurve(QQ, [Integer(1),Integer(1)]) >>> E.gens() [(0 : 1 : 1)]
R.<a1,a2,a3,a4,a6> = QQ[] E = EllipticCurve([a1,a2,a3,a4,a6]) E.gens() E = EllipticCurve(QQ, [1,1]) E.gens()
- hyperelliptic_polynomials()[source]¶
Return a pair of polynomials \(g(x)\), \(h(x)\) such that this elliptic curve can be defined by the standard hyperelliptic equation
\[y^2 + h(x)y = g(x).\]EXAMPLES:
sage: R.<a1,a2,a3,a4,a6>=QQ[] sage: E = EllipticCurve([a1,a2,a3,a4,a6]) sage: E.hyperelliptic_polynomials() (x^3 + a2*x^2 + a4*x + a6, a1*x + a3)
>>> from sage.all import * >>> R = QQ['a1, a2, a3, a4, a6']; (a1, a2, a3, a4, a6,) = R._first_ngens(5) >>> E = EllipticCurve([a1,a2,a3,a4,a6]) >>> E.hyperelliptic_polynomials() (x^3 + a2*x^2 + a4*x + a6, a1*x + a3)
R.<a1,a2,a3,a4,a6>=QQ[] E = EllipticCurve([a1,a2,a3,a4,a6]) E.hyperelliptic_polynomials()
- identity_morphism()[source]¶
Return the identity endomorphism of this elliptic curve as an
EllipticCurveHomobject.EXAMPLES:
sage: E = EllipticCurve(j=42) sage: E.identity_morphism() Elliptic-curve endomorphism of Elliptic Curve defined by y^2 = x^3 + 5901*x + 1105454 over Rational Field Via: (u,r,s,t) = (1, 0, 0, 0) sage: E.identity_morphism() == E.scalar_multiplication(1) True
>>> from sage.all import * >>> E = EllipticCurve(j=Integer(42)) >>> E.identity_morphism() Elliptic-curve endomorphism of Elliptic Curve defined by y^2 = x^3 + 5901*x + 1105454 over Rational Field Via: (u,r,s,t) = (1, 0, 0, 0) >>> E.identity_morphism() == E.scalar_multiplication(Integer(1)) True
E = EllipticCurve(j=42) E.identity_morphism() E.identity_morphism() == E.scalar_multiplication(1)
- is_isomorphic(other, field=None)[source]¶
Return whether or not
selfis isomorphic toother.INPUT:
other– another elliptic curvefield– (default:None) a field into which the coefficients of the curves may be coerced (by default, uses the base field of the curves).
OUTPUT:
boolean;
Trueif there is an isomorphism from curveselfto curveotherdefined overfield.EXAMPLES:
sage: E = EllipticCurve('389a') sage: F = E.change_weierstrass_model([2,3,4,5]); F Elliptic Curve defined by y^2 + 4*x*y + 11/8*y = x^3 - 3/2*x^2 - 13/16*x over Rational Field sage: E.is_isomorphic(F) True sage: E.is_isomorphic(F.change_ring(CC)) False
>>> from sage.all import * >>> E = EllipticCurve('389a') >>> F = E.change_weierstrass_model([Integer(2),Integer(3),Integer(4),Integer(5)]); F Elliptic Curve defined by y^2 + 4*x*y + 11/8*y = x^3 - 3/2*x^2 - 13/16*x over Rational Field >>> E.is_isomorphic(F) True >>> E.is_isomorphic(F.change_ring(CC)) False
E = EllipticCurve('389a') F = E.change_weierstrass_model([2,3,4,5]); F E.is_isomorphic(F) E.is_isomorphic(F.change_ring(CC))
- is_on_curve(x, y)[source]¶
Return
Trueif \((x,y)\) is an affine point on this curve.INPUT:
x,y– elements of the base ring of the curve
EXAMPLES:
sage: E = EllipticCurve(QQ,[1,1]) sage: E.is_on_curve(0,1) True sage: E.is_on_curve(1,1) False
>>> from sage.all import * >>> E = EllipticCurve(QQ,[Integer(1),Integer(1)]) >>> E.is_on_curve(Integer(0),Integer(1)) True >>> E.is_on_curve(Integer(1),Integer(1)) False
E = EllipticCurve(QQ,[1,1]) E.is_on_curve(0,1) E.is_on_curve(1,1)
- is_x_coord(x)[source]¶
Return
Trueifxis the \(x\)-coordinate of a point on this curve.Note
See also
lift_x()to find the point(s) with a given \(x\)-coordinate. This function may be useful in cases where testing an element of the base field for being a square is faster than finding its square root.EXAMPLES:
sage: E = EllipticCurve('37a'); E Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field sage: E.is_x_coord(1) True sage: E.is_x_coord(2) True
>>> from sage.all import * >>> E = EllipticCurve('37a'); E Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field >>> E.is_x_coord(Integer(1)) True >>> E.is_x_coord(Integer(2)) True
E = EllipticCurve('37a'); E E.is_x_coord(1) E.is_x_coord(2)There are no rational points with x-coordinate 3:
sage: E.is_x_coord(3) False
>>> from sage.all import * >>> E.is_x_coord(Integer(3)) False
E.is_x_coord(3)
However, there are such points in \(E(\RR)\):
sage: E.change_ring(RR).is_x_coord(3) True
>>> from sage.all import * >>> E.change_ring(RR).is_x_coord(Integer(3)) True
E.change_ring(RR).is_x_coord(3)
And of course it always works in \(E(\CC)\):
sage: E.change_ring(RR).is_x_coord(-3) False sage: E.change_ring(CC).is_x_coord(-3) True
>>> from sage.all import * >>> E.change_ring(RR).is_x_coord(-Integer(3)) False >>> E.change_ring(CC).is_x_coord(-Integer(3)) True
E.change_ring(RR).is_x_coord(-3) E.change_ring(CC).is_x_coord(-3)
AUTHORS:
John Cremona (2008-08-07): adapted from
lift_x()
- isomorphism(u, r, s=0, t=0, is_codomain=0)[source]¶
Given four values \(u,r,s,t\) in the base ring of this curve, return the
WeierstrassIsomorphismdefined by \(u,r,s,t\) with this curve as its codomain. (The value \(u\) must be a unit; the values \(r,s,t\) default to zero.)Optionally, if the keyword argument
is_codomainis set toTrue, return the isomorphism defined by \(u,r,s,t\) with this curve as its codomain.EXAMPLES:
sage: E = EllipticCurve([1, 2, 3, 4, 5]) sage: iso = E.isomorphism(6); iso Elliptic-curve morphism: From: Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Rational Field To: Elliptic Curve defined by y^2 + 1/6*x*y + 1/72*y = x^3 + 1/18*x^2 + 1/324*x + 5/46656 over Rational Field Via: (u,r,s,t) = (6, 0, 0, 0) sage: iso.domain() == E True sage: iso.codomain() == E.scale_curve(1 / 6) True sage: iso = E.isomorphism(1, 7, 8, 9); iso Elliptic-curve morphism: From: Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Rational Field To: Elliptic Curve defined by y^2 + 17*x*y + 28*y = x^3 - 49*x^2 - 54*x + 303 over Rational Field Via: (u,r,s,t) = (1, 7, 8, 9) sage: iso.domain() == E True sage: iso.codomain() == E.rst_transform(7, 8, 9) True sage: iso = E.isomorphism(6, 7, 8, 9); iso Elliptic-curve morphism: From: Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Rational Field To: Elliptic Curve defined by y^2 + 17/6*x*y + 7/54*y = x^3 - 49/36*x^2 - 1/24*x + 101/15552 over Rational Field Via: (u,r,s,t) = (6, 7, 8, 9) sage: iso.domain() == E True sage: iso.codomain() == E.rst_transform(7, 8, 9).scale_curve(1 / 6) True
>>> from sage.all import * >>> E = EllipticCurve([Integer(1), Integer(2), Integer(3), Integer(4), Integer(5)]) >>> iso = E.isomorphism(Integer(6)); iso Elliptic-curve morphism: From: Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Rational Field To: Elliptic Curve defined by y^2 + 1/6*x*y + 1/72*y = x^3 + 1/18*x^2 + 1/324*x + 5/46656 over Rational Field Via: (u,r,s,t) = (6, 0, 0, 0) >>> iso.domain() == E True >>> iso.codomain() == E.scale_curve(Integer(1) / Integer(6)) True >>> iso = E.isomorphism(Integer(1), Integer(7), Integer(8), Integer(9)); iso Elliptic-curve morphism: From: Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Rational Field To: Elliptic Curve defined by y^2 + 17*x*y + 28*y = x^3 - 49*x^2 - 54*x + 303 over Rational Field Via: (u,r,s,t) = (1, 7, 8, 9) >>> iso.domain() == E True >>> iso.codomain() == E.rst_transform(Integer(7), Integer(8), Integer(9)) True >>> iso = E.isomorphism(Integer(6), Integer(7), Integer(8), Integer(9)); iso Elliptic-curve morphism: From: Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Rational Field To: Elliptic Curve defined by y^2 + 17/6*x*y + 7/54*y = x^3 - 49/36*x^2 - 1/24*x + 101/15552 over Rational Field Via: (u,r,s,t) = (6, 7, 8, 9) >>> iso.domain() == E True >>> iso.codomain() == E.rst_transform(Integer(7), Integer(8), Integer(9)).scale_curve(Integer(1) / Integer(6)) True
E = EllipticCurve([1, 2, 3, 4, 5]) iso = E.isomorphism(6); iso iso.domain() == E iso.codomain() == E.scale_curve(1 / 6) iso = E.isomorphism(1, 7, 8, 9); iso iso.domain() == E iso.codomain() == E.rst_transform(7, 8, 9) iso = E.isomorphism(6, 7, 8, 9); iso iso.domain() == E iso.codomain() == E.rst_transform(7, 8, 9).scale_curve(1 / 6)
The
is_codomainargument reverses the role of domain and codomain:sage: E = EllipticCurve([1, 2, 3, 4, 5]) sage: iso = E.isomorphism(6, is_codomain=True); iso Elliptic-curve morphism: From: Elliptic Curve defined by y^2 + 6*x*y + 648*y = x^3 + 72*x^2 + 5184*x + 233280 over Rational Field To: Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Rational Field Via: (u,r,s,t) = (6, 0, 0, 0) sage: iso.domain() == E.scale_curve(6) True sage: iso.codomain() == E True sage: iso = E.isomorphism(1, 7, 8, 9, is_codomain=True); iso Elliptic-curve morphism: From: Elliptic Curve defined by y^2 - 15*x*y + 90*y = x^3 - 75*x^2 + 796*x - 2289 over Rational Field To: Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Rational Field Via: (u,r,s,t) = (1, 7, 8, 9) sage: iso.domain().rst_transform(7, 8, 9) == E True sage: iso.codomain() == E True sage: iso = E.isomorphism(6, 7, 8, 9, is_codomain=True); iso Elliptic-curve morphism: From: Elliptic Curve defined by y^2 - 10*x*y + 700*y = x^3 + 35*x^2 + 9641*x + 169486 over Rational Field To: Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Rational Field Via: (u,r,s,t) = (6, 7, 8, 9) sage: iso.domain().rst_transform(7, 8, 9) == E.scale_curve(6) True sage: iso.codomain() == E True
>>> from sage.all import * >>> E = EllipticCurve([Integer(1), Integer(2), Integer(3), Integer(4), Integer(5)]) >>> iso = E.isomorphism(Integer(6), is_codomain=True); iso Elliptic-curve morphism: From: Elliptic Curve defined by y^2 + 6*x*y + 648*y = x^3 + 72*x^2 + 5184*x + 233280 over Rational Field To: Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Rational Field Via: (u,r,s,t) = (6, 0, 0, 0) >>> iso.domain() == E.scale_curve(Integer(6)) True >>> iso.codomain() == E True >>> iso = E.isomorphism(Integer(1), Integer(7), Integer(8), Integer(9), is_codomain=True); iso Elliptic-curve morphism: From: Elliptic Curve defined by y^2 - 15*x*y + 90*y = x^3 - 75*x^2 + 796*x - 2289 over Rational Field To: Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Rational Field Via: (u,r,s,t) = (1, 7, 8, 9) >>> iso.domain().rst_transform(Integer(7), Integer(8), Integer(9)) == E True >>> iso.codomain() == E True >>> iso = E.isomorphism(Integer(6), Integer(7), Integer(8), Integer(9), is_codomain=True); iso Elliptic-curve morphism: From: Elliptic Curve defined by y^2 - 10*x*y + 700*y = x^3 + 35*x^2 + 9641*x + 169486 over Rational Field To: Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Rational Field Via: (u,r,s,t) = (6, 7, 8, 9) >>> iso.domain().rst_transform(Integer(7), Integer(8), Integer(9)) == E.scale_curve(Integer(6)) True >>> iso.codomain() == E True
E = EllipticCurve([1, 2, 3, 4, 5]) iso = E.isomorphism(6, is_codomain=True); iso iso.domain() == E.scale_curve(6) iso.codomain() == E iso = E.isomorphism(1, 7, 8, 9, is_codomain=True); iso iso.domain().rst_transform(7, 8, 9) == E iso.codomain() == E iso = E.isomorphism(6, 7, 8, 9, is_codomain=True); iso iso.domain().rst_transform(7, 8, 9) == E.scale_curve(6) iso.codomain() == E
- isomorphism_to(other)[source]¶
Given another weierstrass model
otherofself, return an isomorphism fromselftoother.INPUT:
other– an elliptic curve isomorphic toself
OUTPUT:
(Weierstrassmorphism) An isomorphism from
selftoother.Note
If the curves in question are not isomorphic, a
ValueErroris raised.EXAMPLES:
sage: E = EllipticCurve('37a') sage: F = E.short_weierstrass_model() sage: w = E.isomorphism_to(F); w Elliptic-curve morphism: From: Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field To: Elliptic Curve defined by y^2 = x^3 - 16*x + 16 over Rational Field Via: (u,r,s,t) = (1/2, 0, 0, -1/2) sage: P = E(0,-1,1) sage: w(P) (0 : -4 : 1) sage: w(5*P) (1 : 1 : 1) sage: 5*w(P) (1 : 1 : 1) sage: 120*w(P) == w(120*P) True
>>> from sage.all import * >>> E = EllipticCurve('37a') >>> F = E.short_weierstrass_model() >>> w = E.isomorphism_to(F); w Elliptic-curve morphism: From: Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field To: Elliptic Curve defined by y^2 = x^3 - 16*x + 16 over Rational Field Via: (u,r,s,t) = (1/2, 0, 0, -1/2) >>> P = E(Integer(0),-Integer(1),Integer(1)) >>> w(P) (0 : -4 : 1) >>> w(Integer(5)*P) (1 : 1 : 1) >>> Integer(5)*w(P) (1 : 1 : 1) >>> Integer(120)*w(P) == w(Integer(120)*P) True
E = EllipticCurve('37a') F = E.short_weierstrass_model() w = E.isomorphism_to(F); w P = E(0,-1,1) w(P) w(5*P) 5*w(P) 120*w(P) == w(120*P)We can also handle injections to different base rings:
sage: x = polygen(ZZ, 'x') sage: K.<a> = NumberField(x^3 - 7) # needs sage.rings.number_field sage: E.isomorphism_to(E.change_ring(K)) # needs sage.rings.number_field Elliptic-curve morphism: From: Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field To: Elliptic Curve defined by y^2 + y = x^3 + (-1)*x over Number Field in a with defining polynomial x^3 - 7 Via: (u,r,s,t) = (1, 0, 0, 0)
>>> from sage.all import * >>> x = polygen(ZZ, 'x') >>> K = NumberField(x**Integer(3) - Integer(7), names=('a',)); (a,) = K._first_ngens(1)# needs sage.rings.number_field >>> E.isomorphism_to(E.change_ring(K)) # needs sage.rings.number_field Elliptic-curve morphism: From: Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field To: Elliptic Curve defined by y^2 + y = x^3 + (-1)*x over Number Field in a with defining polynomial x^3 - 7 Via: (u,r,s,t) = (1, 0, 0, 0)
x = polygen(ZZ, 'x') K.<a> = NumberField(x^3 - 7) # needs sage.rings.number_field E.isomorphism_to(E.change_ring(K)) # needs sage.rings.number_field
- isomorphisms(other, field=None)[source]¶
Return the set of isomorphisms from
selftoother(as a list).INPUT:
other– another elliptic curvefield– (default:None) a field into which the coefficients of the curves may be coerced (by default, uses the base field of the curves)
OUTPUT:
A list of
WeierstrassIsomorphismobjects consisting of all the isomorphisms from the curveselfto the curveotherdefined overfield.EXAMPLES:
sage: E = EllipticCurve_from_j(QQ(0)) # a curve with j=0 over QQ sage: F = EllipticCurve('27a3') # should be the same one sage: E.isomorphisms(F) [Elliptic-curve endomorphism of Elliptic Curve defined by y^2 + y = x^3 over Rational Field Via: (u,r,s,t) = (1, 0, 0, 0), Elliptic-curve endomorphism of Elliptic Curve defined by y^2 + y = x^3 over Rational Field Via: (u,r,s,t) = (-1, 0, 0, -1)]
>>> from sage.all import * >>> E = EllipticCurve_from_j(QQ(Integer(0))) # a curve with j=0 over QQ >>> F = EllipticCurve('27a3') # should be the same one >>> E.isomorphisms(F) [Elliptic-curve endomorphism of Elliptic Curve defined by y^2 + y = x^3 over Rational Field Via: (u,r,s,t) = (1, 0, 0, 0), Elliptic-curve endomorphism of Elliptic Curve defined by y^2 + y = x^3 over Rational Field Via: (u,r,s,t) = (-1, 0, 0, -1)]
E = EllipticCurve_from_j(QQ(0)) # a curve with j=0 over QQ F = EllipticCurve('27a3') # should be the same one E.isomorphisms(F)We can also find isomorphisms defined over extension fields:
sage: # needs sage.rings.finite_rings sage: E = EllipticCurve(GF(7), [0,0,0,1,1]) sage: F = EllipticCurve(GF(7), [0,0,0,1,-1]) sage: E.isomorphisms(F) [] sage: E.isomorphisms(F, GF(49,'a')) [Elliptic-curve morphism: From: Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field in a of size 7^2 To: Elliptic Curve defined by y^2 = x^3 + x + 6 over Finite Field in a of size 7^2 Via: (u,r,s,t) = (a + 3, 0, 0, 0), Elliptic-curve morphism: From: Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field in a of size 7^2 To: Elliptic Curve defined by y^2 = x^3 + x + 6 over Finite Field in a of size 7^2 Via: (u,r,s,t) = (6*a + 4, 0, 0, 0)]
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> E = EllipticCurve(GF(Integer(7)), [Integer(0),Integer(0),Integer(0),Integer(1),Integer(1)]) >>> F = EllipticCurve(GF(Integer(7)), [Integer(0),Integer(0),Integer(0),Integer(1),-Integer(1)]) >>> E.isomorphisms(F) [] >>> E.isomorphisms(F, GF(Integer(49),'a')) [Elliptic-curve morphism: From: Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field in a of size 7^2 To: Elliptic Curve defined by y^2 = x^3 + x + 6 over Finite Field in a of size 7^2 Via: (u,r,s,t) = (a + 3, 0, 0, 0), Elliptic-curve morphism: From: Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field in a of size 7^2 To: Elliptic Curve defined by y^2 = x^3 + x + 6 over Finite Field in a of size 7^2 Via: (u,r,s,t) = (6*a + 4, 0, 0, 0)]
# needs sage.rings.finite_rings E = EllipticCurve(GF(7), [0,0,0,1,1]) F = EllipticCurve(GF(7), [0,0,0,1,-1]) E.isomorphisms(F) E.isomorphisms(F, GF(49,'a'))
- j_invariant()[source]¶
Return the \(j\)-invariant of this elliptic curve.
This method is cached.
EXAMPLES:
sage: E = EllipticCurve([0,0,1,-1,0]) sage: E.j_invariant() 110592/37 sage: E = EllipticCurve([0, -1, 1, -10, -20]) sage: E.j_invariant() -122023936/161051 sage: E = EllipticCurve([-4,0]) sage: E.j_invariant() 1728 sage: E = EllipticCurve([GF(7)(2),1]) sage: E.j_invariant() 1
>>> from sage.all import * >>> E = EllipticCurve([Integer(0),Integer(0),Integer(1),-Integer(1),Integer(0)]) >>> E.j_invariant() 110592/37 >>> E = EllipticCurve([Integer(0), -Integer(1), Integer(1), -Integer(10), -Integer(20)]) >>> E.j_invariant() -122023936/161051 >>> E = EllipticCurve([-Integer(4),Integer(0)]) >>> E.j_invariant() 1728 >>> E = EllipticCurve([GF(Integer(7))(Integer(2)),Integer(1)]) >>> E.j_invariant() 1
E = EllipticCurve([0,0,1,-1,0]) E.j_invariant() E = EllipticCurve([0, -1, 1, -10, -20]) E.j_invariant() E = EllipticCurve([-4,0]) E.j_invariant() E = EllipticCurve([GF(7)(2),1]) E.j_invariant()
- lift_x(x, all=False, extend=False)[source]¶
Return one or all points with given \(x\)-coordinate.
This method is deterministic: It returns the same data each time when called again with the same \(x\).
INPUT:
x– an element of the base ring of the curve, or of an extensionall– boolean (default:False); ifTrue, return a (possibly empty) list of all points; ifFalse, return just one point, or raise aValueErrorif there are none.extend– boolean (default:False);if
False, extend the base if necessary and possible to include \(x\), and only return point(s) defined over this ring, or raise an error when there are none with this \(x\)-coordinate;If
True, the base ring will be extended if necessary to contain the \(y\)-coordinates of the point(s) with this \(x\)-coordinate, in addition to a possible base change to include \(x\).
OUTPUT:
A point or list of up to 2 points on this curve, or a base-change of this curve to a larger ring.
See also
EXAMPLES:
sage: E = EllipticCurve('37a'); E Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field sage: E.lift_x(1) (1 : -1 : 1) sage: E.lift_x(2) (2 : -3 : 1) sage: E.lift_x(1/4, all=True) [(1/4 : -5/8 : 1), (1/4 : -3/8 : 1)]
>>> from sage.all import * >>> E = EllipticCurve('37a'); E Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field >>> E.lift_x(Integer(1)) (1 : -1 : 1) >>> E.lift_x(Integer(2)) (2 : -3 : 1) >>> E.lift_x(Integer(1)/Integer(4), all=True) [(1/4 : -5/8 : 1), (1/4 : -3/8 : 1)]
E = EllipticCurve('37a'); E E.lift_x(1) E.lift_x(2) E.lift_x(1/4, all=True)There are no rational points with \(x\)-coordinate 3:
sage: E.lift_x(3) Traceback (most recent call last): ... ValueError: No point with x-coordinate 3 on Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
>>> from sage.all import * >>> E.lift_x(Integer(3)) Traceback (most recent call last): ... ValueError: No point with x-coordinate 3 on Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
E.lift_x(3)
We can use the
extendparameter to make the necessary quadratic extension. Note that in such cases the returned point is a point on a new curve object, the result of changing the base ring to the parent of \(x\):sage: P = E.lift_x(3, extend=True); P # needs sage.rings.number_field (3 : -y - 1 : 1) sage: P.curve() # needs sage.rings.number_field Elliptic Curve defined by y^2 + y = x^3 + (-1)*x over Number Field in y with defining polynomial y^2 + y - 24
>>> from sage.all import * >>> P = E.lift_x(Integer(3), extend=True); P # needs sage.rings.number_field (3 : -y - 1 : 1) >>> P.curve() # needs sage.rings.number_field Elliptic Curve defined by y^2 + y = x^3 + (-1)*x over Number Field in y with defining polynomial y^2 + y - 24
P = E.lift_x(3, extend=True); P # needs sage.rings.number_field P.curve() # needs sage.rings.number_field
Or we can extend scalars. There are two such points in \(E(\RR)\):
sage: E.change_ring(RR).lift_x(3, all=True) [(3.00000000000000 : -5.42442890089805 : 1.00000000000000), (3.00000000000000 : 4.42442890089805 : 1.00000000000000)]
>>> from sage.all import * >>> E.change_ring(RR).lift_x(Integer(3), all=True) [(3.00000000000000 : -5.42442890089805 : 1.00000000000000), (3.00000000000000 : 4.42442890089805 : 1.00000000000000)]
E.change_ring(RR).lift_x(3, all=True)
And of course it always works in \(E(\CC)\):
sage: E.change_ring(RR).lift_x(.5, all=True) [] sage: E.change_ring(CC).lift_x(.5) (0.500000000000000 : -0.500000000000000 - 0.353553390593274*I : 1.00000000000000)
>>> from sage.all import * >>> E.change_ring(RR).lift_x(RealNumber('.5'), all=True) [] >>> E.change_ring(CC).lift_x(RealNumber('.5')) (0.500000000000000 : -0.500000000000000 - 0.353553390593274*I : 1.00000000000000)
E.change_ring(RR).lift_x(.5, all=True) E.change_ring(CC).lift_x(.5)
In this example we start with a curve defined over \(\QQ\) which has no rational points with \(x=0\), but using
extend = Truewe can construct such a point over a quadratic field:sage: E = EllipticCurve([0,0,0,0,2]); E Elliptic Curve defined by y^2 = x^3 + 2 over Rational Field sage: P = E.lift_x(0, extend=True); P # needs sage.rings.number_field (0 : -y : 1) sage: P.curve() # needs sage.rings.number_field Elliptic Curve defined by y^2 = x^3 + 2 over Number Field in y with defining polynomial y^2 - 2
>>> from sage.all import * >>> E = EllipticCurve([Integer(0),Integer(0),Integer(0),Integer(0),Integer(2)]); E Elliptic Curve defined by y^2 = x^3 + 2 over Rational Field >>> P = E.lift_x(Integer(0), extend=True); P # needs sage.rings.number_field (0 : -y : 1) >>> P.curve() # needs sage.rings.number_field Elliptic Curve defined by y^2 = x^3 + 2 over Number Field in y with defining polynomial y^2 - 2
E = EllipticCurve([0,0,0,0,2]); E P = E.lift_x(0, extend=True); P # needs sage.rings.number_field P.curve() # needs sage.rings.number_field
We can perform these operations over finite fields too:
sage: E = EllipticCurve('37a').change_ring(GF(17)); E Elliptic Curve defined by y^2 + y = x^3 + 16*x over Finite Field of size 17 sage: E.lift_x(7) (7 : 5 : 1) sage: E.lift_x(3) Traceback (most recent call last): ... ValueError: No point with x-coordinate 3 on Elliptic Curve defined by y^2 + y = x^3 + 16*x over Finite Field of size 17
>>> from sage.all import * >>> E = EllipticCurve('37a').change_ring(GF(Integer(17))); E Elliptic Curve defined by y^2 + y = x^3 + 16*x over Finite Field of size 17 >>> E.lift_x(Integer(7)) (7 : 5 : 1) >>> E.lift_x(Integer(3)) Traceback (most recent call last): ... ValueError: No point with x-coordinate 3 on Elliptic Curve defined by y^2 + y = x^3 + 16*x over Finite Field of size 17
E = EllipticCurve('37a').change_ring(GF(17)); E E.lift_x(7) E.lift_x(3)Note that there is only one lift with \(x\)-coordinate 10 in \(E(\GF{17})\):
sage: E.lift_x(10, all=True) [(10 : 8 : 1)]
>>> from sage.all import * >>> E.lift_x(Integer(10), all=True) [(10 : 8 : 1)]
E.lift_x(10, all=True)
We can lift over more exotic rings too. If the supplied x value is in an extension of the base, note that the point returned is on the base-extended curve:
sage: E = EllipticCurve('37a') sage: P = E.lift_x(pAdicField(17, 5)(6)); P # needs sage.rings.padics (6 + O(17^5) : 14 + O(17^5) : 1 + O(17^5)) sage: P.curve() # needs sage.rings.padics Elliptic Curve defined by y^2 + (1+O(17^5))*y = x^3 + (16+16*17+16*17^2+16*17^3+16*17^4+O(17^5))*x over 17-adic Field with capped relative precision 5 sage: K.<t> = PowerSeriesRing(QQ, 't', 5) sage: P = E.lift_x(1 + t); P (1 + t : -1 - 2*t + t^2 - 5*t^3 + 21*t^4 + O(t^5) : 1) sage: K.<a> = GF(16) # needs sage.rings.finite_rings sage: P = E.change_ring(K).lift_x(a^3); P # needs sage.rings.finite_rings (a^3 : a^3 + a : 1) sage: P.curve() # needs sage.rings.finite_rings Elliptic Curve defined by y^2 + y = x^3 + x over Finite Field in a of size 2^4
>>> from sage.all import * >>> E = EllipticCurve('37a') >>> P = E.lift_x(pAdicField(Integer(17), Integer(5))(Integer(6))); P # needs sage.rings.padics (6 + O(17^5) : 14 + O(17^5) : 1 + O(17^5)) >>> P.curve() # needs sage.rings.padics Elliptic Curve defined by y^2 + (1+O(17^5))*y = x^3 + (16+16*17+16*17^2+16*17^3+16*17^4+O(17^5))*x over 17-adic Field with capped relative precision 5 >>> K = PowerSeriesRing(QQ, 't', Integer(5), names=('t',)); (t,) = K._first_ngens(1) >>> P = E.lift_x(Integer(1) + t); P (1 + t : -1 - 2*t + t^2 - 5*t^3 + 21*t^4 + O(t^5) : 1) >>> K = GF(Integer(16), names=('a',)); (a,) = K._first_ngens(1)# needs sage.rings.finite_rings >>> P = E.change_ring(K).lift_x(a**Integer(3)); P # needs sage.rings.finite_rings (a^3 : a^3 + a : 1) >>> P.curve() # needs sage.rings.finite_rings Elliptic Curve defined by y^2 + y = x^3 + x over Finite Field in a of size 2^4
E = EllipticCurve('37a') P = E.lift_x(pAdicField(17, 5)(6)); P # needs sage.rings.padics P.curve() # needs sage.rings.padics K.<t> = PowerSeriesRing(QQ, 't', 5) P = E.lift_x(1 + t); P K.<a> = GF(16) # needs sage.rings.finite_rings P = E.change_ring(K).lift_x(a^3); P # needs sage.rings.finite_rings P.curve() # needs sage.rings.finite_ringsWe can extend the base field to include the associated \(y\) value(s):
sage: E = EllipticCurve([0,0,0,0,2]); E Elliptic Curve defined by y^2 = x^3 + 2 over Rational Field sage: x = polygen(QQ) sage: P = E.lift_x(x, extend=True); P (x : -y : 1)
>>> from sage.all import * >>> E = EllipticCurve([Integer(0),Integer(0),Integer(0),Integer(0),Integer(2)]); E Elliptic Curve defined by y^2 = x^3 + 2 over Rational Field >>> x = polygen(QQ) >>> P = E.lift_x(x, extend=True); P (x : -y : 1)
E = EllipticCurve([0,0,0,0,2]); E x = polygen(QQ) P = E.lift_x(x, extend=True); P
This point is a generic point on E:
sage: P.curve() Elliptic Curve defined by y^2 = x^3 + 2 over Univariate Quotient Polynomial Ring in y over Fraction Field of Univariate Polynomial Ring in x over Rational Field with modulus y^2 - x^3 - 2 sage: -P (x : y : 1) sage: 2*P ((1/4*x^4 - 4*x)/(x^3 + 2) : ((-1/8*x^6 - 5*x^3 + 4)/(x^6 + 4*x^3 + 4))*y : 1)
>>> from sage.all import * >>> P.curve() Elliptic Curve defined by y^2 = x^3 + 2 over Univariate Quotient Polynomial Ring in y over Fraction Field of Univariate Polynomial Ring in x over Rational Field with modulus y^2 - x^3 - 2 >>> -P (x : y : 1) >>> Integer(2)*P ((1/4*x^4 - 4*x)/(x^3 + 2) : ((-1/8*x^6 - 5*x^3 + 4)/(x^6 + 4*x^3 + 4))*y : 1)
P.curve() -P 2*P
Check that Issue #30297 is fixed:
sage: K = Qp(5) # needs sage.rings.padics sage: E = EllipticCurve([K(0), K(1)]) # needs sage.rings.padics sage: E.lift_x(1, extend=True) # needs sage.rings.padics (1 + O(5^20) : y + O(5^20) : 1 + O(5^20))
>>> from sage.all import * >>> K = Qp(Integer(5)) # needs sage.rings.padics >>> E = EllipticCurve([K(Integer(0)), K(Integer(1))]) # needs sage.rings.padics >>> E.lift_x(Integer(1), extend=True) # needs sage.rings.padics (1 + O(5^20) : y + O(5^20) : 1 + O(5^20))
K = Qp(5) # needs sage.rings.padics E = EllipticCurve([K(0), K(1)]) # needs sage.rings.padics E.lift_x(1, extend=True) # needs sage.rings.padics
AUTHORS:
Robert Bradshaw (2007-04-24)
John Cremona (2017-11-10)
- montgomery_model(twisted=False, morphism=False)[source]¶
Return a (twisted or untwisted) Montgomery model for this elliptic curve, if possible.
A Montgomery curve is a smooth projective curve of the form
\[BY^2 = X^3 + AX^2 + X.\]The Montgomery curve is called untwisted if \(B=1\).
INPUT:
twisted– boolean (default:False); allow \(B \neq 1\)morphism– boolean (default:False); also return an isomorphism from this curve to the computed Montgomery model
OUTPUT:
If
twistedisFalse(the default), anEllipticCurve_genericobject encapsulating an untwisted Montgomery curve. Otherwise, aProjectivePlaneCurveobject encapsulating a (potentially twisted) Montgomery curve.If
morphismisTrue, this method returns a tuple consisting of such a curve together with an isomorphism of suitable type (eitherWeierstrassIsomorphismorWeierstrassTransformationWithInverse) from this curve to the Montgomery model.EXAMPLES:
sage: E = EllipticCurve(QQbar, '11a1') # needs sage.rings.number_field sage: E.montgomery_model() # needs sage.rings.number_field Elliptic Curve defined by y^2 = x^3 + (-1.953522420987248?)*x^2 + x over Algebraic Field
>>> from sage.all import * >>> E = EllipticCurve(QQbar, '11a1') # needs sage.rings.number_field >>> E.montgomery_model() # needs sage.rings.number_field Elliptic Curve defined by y^2 = x^3 + (-1.953522420987248?)*x^2 + x over Algebraic Field
E = EllipticCurve(QQbar, '11a1') # needs sage.rings.number_field E.montgomery_model() # needs sage.rings.number_field
sage: E = EllipticCurve(GF(431^2), [7,7]) # needs sage.rings.finite_rings sage: E.montgomery_model() # needs sage.rings.finite_rings Elliptic Curve defined by y^2 = x^3 + (51*z2+190)*x^2 + x over Finite Field in z2 of size 431^2
>>> from sage.all import * >>> E = EllipticCurve(GF(Integer(431)**Integer(2)), [Integer(7),Integer(7)]) # needs sage.rings.finite_rings >>> E.montgomery_model() # needs sage.rings.finite_rings Elliptic Curve defined by y^2 = x^3 + (51*z2+190)*x^2 + x over Finite Field in z2 of size 431^2
E = EllipticCurve(GF(431^2), [7,7]) # needs sage.rings.finite_rings E.montgomery_model() # needs sage.rings.finite_rings
>>> from sage.all import * >>> E = EllipticCurve(GF(Integer(431)**Integer(2)), [Integer(7),Integer(7)]) # needs sage.rings.finite_rings >>> E.montgomery_model() # needs sage.rings.finite_rings Elliptic Curve defined by y^2 = x^3 + (51*z2+190)*x^2 + x over Finite Field in z2 of size 431^2
E = EllipticCurve(GF(431^2), [7,7]) # needs sage.rings.finite_rings E.montgomery_model() # needs sage.rings.finite_rings
An isomorphism between the Montgomery and Weierstrass form can be obtained using the
morphismparameter:sage: E.montgomery_model(morphism=True) # needs sage.rings.finite_rings (Elliptic Curve defined by y^2 = x^3 + (51*z2+190)*x^2 + x over Finite Field in z2 of size 431^2, Elliptic-curve morphism: From: Elliptic Curve defined by y^2 = x^3 + 7*x + 7 over Finite Field in z2 of size 431^2 To: Elliptic Curve defined by y^2 = x^3 + (51*z2+190)*x^2 + x over Finite Field in z2 of size 431^2 Via: (u,r,s,t) = (64*z2 + 407, 159, 0, 0))
>>> from sage.all import * >>> E.montgomery_model(morphism=True) # needs sage.rings.finite_rings (Elliptic Curve defined by y^2 = x^3 + (51*z2+190)*x^2 + x over Finite Field in z2 of size 431^2, Elliptic-curve morphism: From: Elliptic Curve defined by y^2 = x^3 + 7*x + 7 over Finite Field in z2 of size 431^2 To: Elliptic Curve defined by y^2 = x^3 + (51*z2+190)*x^2 + x over Finite Field in z2 of size 431^2 Via: (u,r,s,t) = (64*z2 + 407, 159, 0, 0))
E.montgomery_model(morphism=True) # needs sage.rings.finite_rings
Not all elliptic curves have a Montgomery model over their field of definition:
sage: E = EllipticCurve(GF(257), [1,1]) sage: E.montgomery_model() Traceback (most recent call last): ... ValueError: Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field of size 257 has no Montgomery model
>>> from sage.all import * >>> E = EllipticCurve(GF(Integer(257)), [Integer(1),Integer(1)]) >>> E.montgomery_model() Traceback (most recent call last): ... ValueError: Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field of size 257 has no Montgomery model
E = EllipticCurve(GF(257), [1,1]) E.montgomery_model()
sage: E = EllipticCurve(GF(257), [10,10]) sage: E.montgomery_model() Traceback (most recent call last): ... ValueError: Elliptic Curve defined by y^2 = x^3 + 10*x + 10 over Finite Field of size 257 has no untwisted Montgomery model
>>> from sage.all import * >>> E = EllipticCurve(GF(Integer(257)), [Integer(10),Integer(10)]) >>> E.montgomery_model() Traceback (most recent call last): ... ValueError: Elliptic Curve defined by y^2 = x^3 + 10*x + 10 over Finite Field of size 257 has no untwisted Montgomery model
E = EllipticCurve(GF(257), [10,10]) E.montgomery_model()
>>> from sage.all import * >>> E = EllipticCurve(GF(Integer(257)), [Integer(10),Integer(10)]) >>> E.montgomery_model() Traceback (most recent call last): ... ValueError: Elliptic Curve defined by y^2 = x^3 + 10*x + 10 over Finite Field of size 257 has no untwisted Montgomery model
E = EllipticCurve(GF(257), [10,10]) E.montgomery_model()
However, as hinted by the error message, the latter curve does admit a twisted Montgomery model, which can be computed by passing
twisted=True:sage: E.montgomery_model(twisted=True) Projective Plane Curve over Finite Field of size 257 defined by -x^3 + 8*x^2*z - 127*y^2*z - x*z^2
>>> from sage.all import * >>> E.montgomery_model(twisted=True) Projective Plane Curve over Finite Field of size 257 defined by -x^3 + 8*x^2*z - 127*y^2*z - x*z^2
E.montgomery_model(twisted=True)
Since Sage internally represents elliptic curves as (long) Weierstrass curves, which do not feature the Montgomery \(B\) coefficient, the returned curve in this case is merely a
ProjectivePlaneCurverather than the usualEllipticCurve_generic.Arithmetic on curves of this type is not implemented natively, but can easily be emulated by mapping back and forth to the corresponding Weierstrass curve:
sage: C, f = E.montgomery_model(twisted=True, morphism=True) sage: f Scheme morphism: From: Elliptic Curve defined by y^2 = x^3 + 10*x + 10 over Finite Field of size 257 To: Projective Plane Curve over Finite Field of size 257 defined by -x^3 + 8*x^2*z - 127*y^2*z - x*z^2 Defn: Defined on coordinates by sending (x : y : z) to (x + 116*z : -y : -85*z) sage: g = f.inverse(); g Scheme morphism: From: Projective Plane Curve over Finite Field of size 257 defined by -x^3 + 8*x^2*z - 127*y^2*z - x*z^2 To: Elliptic Curve defined by y^2 = x^3 + 10*x + 10 over Finite Field of size 257 Defn: Defined on coordinates by sending (x : y : z) to (-85*x - 116*z : 85*y : z) sage: P = C(70, 8) sage: Q = C(17, 17) sage: P + Q # this doesn't work... Traceback (most recent call last): ... TypeError: unsupported operand parent(s) for +: ... sage: f(g(P) + g(Q)) # ...but this does (107 : 168 : 1)
>>> from sage.all import * >>> C, f = E.montgomery_model(twisted=True, morphism=True) >>> f Scheme morphism: From: Elliptic Curve defined by y^2 = x^3 + 10*x + 10 over Finite Field of size 257 To: Projective Plane Curve over Finite Field of size 257 defined by -x^3 + 8*x^2*z - 127*y^2*z - x*z^2 Defn: Defined on coordinates by sending (x : y : z) to (x + 116*z : -y : -85*z) >>> g = f.inverse(); g Scheme morphism: From: Projective Plane Curve over Finite Field of size 257 defined by -x^3 + 8*x^2*z - 127*y^2*z - x*z^2 To: Elliptic Curve defined by y^2 = x^3 + 10*x + 10 over Finite Field of size 257 Defn: Defined on coordinates by sending (x : y : z) to (-85*x - 116*z : 85*y : z) >>> P = C(Integer(70), Integer(8)) >>> Q = C(Integer(17), Integer(17)) >>> P + Q # this doesn't work... Traceback (most recent call last): ... TypeError: unsupported operand parent(s) for +: ... >>> f(g(P) + g(Q)) # ...but this does (107 : 168 : 1)
C, f = E.montgomery_model(twisted=True, morphism=True) f g = f.inverse(); g P = C(70, 8) Q = C(17, 17) P + Q # this doesn't work... f(g(P) + g(Q)) # ...but this does
Using the fact that the Weil pairing satisfies \(e(\psi(P),\psi(Q)) = e(P,Q)^{\deg\psi}\), even pairings can be emulated in this way (note that isomorphisms have degree \(1\)):
sage: # needs sage.rings.finite_rings sage: F.<z2> = GF(257^2) sage: C_ = C.change_ring(F) sage: g_ = g.change_ring(F) sage: g_(P).order() 12 sage: T = C_(-7 * z2 - 57, 31 * z2 - 52, 1) sage: g_(T).order() 12 sage: g_(P).weil_pairing(g_(T), 12) 15*z2 + 204
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> F = GF(Integer(257)**Integer(2), names=('z2',)); (z2,) = F._first_ngens(1) >>> C_ = C.change_ring(F) >>> g_ = g.change_ring(F) >>> g_(P).order() 12 >>> T = C_(-Integer(7) * z2 - Integer(57), Integer(31) * z2 - Integer(52), Integer(1)) >>> g_(T).order() 12 >>> g_(P).weil_pairing(g_(T), Integer(12)) 15*z2 + 204
# needs sage.rings.finite_rings F.<z2> = GF(257^2) C_ = C.change_ring(F) g_ = g.change_ring(F) g_(P).order() T = C_(-7 * z2 - 57, 31 * z2 - 52, 1) g_(T).order() g_(P).weil_pairing(g_(T), 12)
Another alternative is to simply extend the base field enough for the curve to have an untwisted Montgomery model:
sage: C_ = E.change_ring(F).montgomery_model(); C_ # needs sage.rings.finite_rings Elliptic Curve defined by y^2 = x^3 + 249*x^2 + x over Finite Field in z2 of size 257^2 sage: h = C.defining_polynomial().change_ring(F); h # needs sage.rings.finite_rings -x^3 + 8*x^2*z - 127*y^2*z - x*z^2 sage: C_.is_isomorphic(EllipticCurve_from_cubic(h).codomain()) # needs sage.rings.finite_rings True
>>> from sage.all import * >>> C_ = E.change_ring(F).montgomery_model(); C_ # needs sage.rings.finite_rings Elliptic Curve defined by y^2 = x^3 + 249*x^2 + x over Finite Field in z2 of size 257^2 >>> h = C.defining_polynomial().change_ring(F); h # needs sage.rings.finite_rings -x^3 + 8*x^2*z - 127*y^2*z - x*z^2 >>> C_.is_isomorphic(EllipticCurve_from_cubic(h).codomain()) # needs sage.rings.finite_rings True
C_ = E.change_ring(F).montgomery_model(); C_ # needs sage.rings.finite_rings h = C.defining_polynomial().change_ring(F); h # needs sage.rings.finite_rings C_.is_isomorphic(EllipticCurve_from_cubic(h).codomain()) # needs sage.rings.finite_rings
See also
The inverse conversion — computing a Weierstrass model for a given Montgomery curve — can be performed using
EllipticCurve_from_cubic().ALGORITHM: [CS2018], §2.4
REFERENCES:
Original publication: [Mont1987], §10.3.1
More recent survey article: [CS2018]
- multiplication_by_m(m, x_only=False)[source]¶
Return the multiplication-by-\(m\) map from
selftoself.The result is a pair of rational functions in two variables \(x\), \(y\) (or a rational function in one variable \(x\) if
x_onlyisTrue).INPUT:
m– nonzero integerx_only– boolean (default:False); ifTrue, return only the \(x\)-coordinate of the map (as a rational function in one variable)
OUTPUT:
a pair \((f(x), g(x,y))\), where \(f\) and \(g\) are rational functions with the degree of \(y\) in \(g(x,y)\) exactly 1,
or just \(f(x)\) if
x_onlyisTrue
Note
The result is not cached.
mis allowed to be negative (but not 0).
EXAMPLES:
sage: E = EllipticCurve([-1,3])
>>> from sage.all import * >>> E = EllipticCurve([-Integer(1),Integer(3)])
E = EllipticCurve([-1,3])
We verify that multiplication by 1 is just the identity:
sage: E.multiplication_by_m(1) (x, y)
>>> from sage.all import * >>> E.multiplication_by_m(Integer(1)) (x, y)
E.multiplication_by_m(1)
Multiplication by 2 is more complicated:
sage: f = E.multiplication_by_m(2) sage: f ((x^4 + 2*x^2 - 24*x + 1)/(4*x^3 - 4*x + 12), (8*x^6*y - 40*x^4*y + 480*x^3*y - 40*x^2*y + 96*x*y - 568*y)/(64*x^6 - 128*x^4 + 384*x^3 + 64*x^2 - 384*x + 576))
>>> from sage.all import * >>> f = E.multiplication_by_m(Integer(2)) >>> f ((x^4 + 2*x^2 - 24*x + 1)/(4*x^3 - 4*x + 12), (8*x^6*y - 40*x^4*y + 480*x^3*y - 40*x^2*y + 96*x*y - 568*y)/(64*x^6 - 128*x^4 + 384*x^3 + 64*x^2 - 384*x + 576))
f = E.multiplication_by_m(2) f
Grab only the x-coordinate (less work):
sage: mx = E.multiplication_by_m(2, x_only=True); mx (1/4*x^4 + 1/2*x^2 - 6*x + 1/4)/(x^3 - x + 3) sage: mx.parent() Fraction Field of Univariate Polynomial Ring in x over Rational Field
>>> from sage.all import * >>> mx = E.multiplication_by_m(Integer(2), x_only=True); mx (1/4*x^4 + 1/2*x^2 - 6*x + 1/4)/(x^3 - x + 3) >>> mx.parent() Fraction Field of Univariate Polynomial Ring in x over Rational Field
mx = E.multiplication_by_m(2, x_only=True); mx mx.parent()
We check that it works on a point:
sage: P = E([2,3]) sage: eval = lambda f,P: [fi(P[0],P[1]) for fi in f] sage: assert E(eval(f,P)) == 2*P
>>> from sage.all import * >>> P = E([Integer(2),Integer(3)]) >>> eval = lambda f,P: [fi(P[Integer(0)],P[Integer(1)]) for fi in f] >>> assert E(eval(f,P)) == Integer(2)*P
P = E([2,3]) eval = lambda f,P: [fi(P[0],P[1]) for fi in f] assert E(eval(f,P)) == 2*P
We do the same but with multiplication by 3:
sage: f = E.multiplication_by_m(3) sage: assert E(eval(f,P)) == 3*P
>>> from sage.all import * >>> f = E.multiplication_by_m(Integer(3)) >>> assert E(eval(f,P)) == Integer(3)*P
f = E.multiplication_by_m(3) assert E(eval(f,P)) == 3*P
And the same with multiplication by 4:
sage: f = E.multiplication_by_m(4) sage: assert E(eval(f,P)) == 4*P
>>> from sage.all import * >>> f = E.multiplication_by_m(Integer(4)) >>> assert E(eval(f,P)) == Integer(4)*P
f = E.multiplication_by_m(4) assert E(eval(f,P)) == 4*P
And the same with multiplication by -1,-2,-3,-4:
sage: for m in [-1,-2,-3,-4]: ....: f = E.multiplication_by_m(m) ....: assert E(eval(f,P)) == m*P
>>> from sage.all import * >>> for m in [-Integer(1),-Integer(2),-Integer(3),-Integer(4)]: ... f = E.multiplication_by_m(m) ... assert E(eval(f,P)) == m*P
for m in [-1,-2,-3,-4]: f = E.multiplication_by_m(m) assert E(eval(f,P)) == m*P
- pari_curve()[source]¶
Return the PARI curve corresponding to this elliptic curve.
The result is cached.
EXAMPLES:
sage: # needs sage.libs.pari sage: E = EllipticCurve([RR(0), RR(0), RR(1), RR(-1), RR(0)]) sage: e = E.pari_curve() sage: type(e) <... 'cypari2.gen.Gen'> sage: e.type() 't_VEC' sage: e.disc() 37.0000000000000
>>> from sage.all import * >>> # needs sage.libs.pari >>> E = EllipticCurve([RR(Integer(0)), RR(Integer(0)), RR(Integer(1)), RR(-Integer(1)), RR(Integer(0))]) >>> e = E.pari_curve() >>> type(e) <... 'cypari2.gen.Gen'> >>> e.type() 't_VEC' >>> e.disc() 37.0000000000000
# needs sage.libs.pari E = EllipticCurve([RR(0), RR(0), RR(1), RR(-1), RR(0)]) e = E.pari_curve() type(e) e.type() e.disc()
Over a finite field:
sage: EllipticCurve(GF(41), [2,5]).pari_curve() # needs sage.libs.pari [Mod(0, 41), Mod(0, 41), Mod(0, 41), Mod(2, 41), Mod(5, 41), Mod(0, 41), Mod(4, 41), Mod(20, 41), Mod(37, 41), Mod(27, 41), Mod(26, 41), Mod(4, 41), Mod(11, 41), Vecsmall([3]), [41, [9, 31, [6, 0, 0, 0]]], [0, 0, 0, 0]]
>>> from sage.all import * >>> EllipticCurve(GF(Integer(41)), [Integer(2),Integer(5)]).pari_curve() # needs sage.libs.pari [Mod(0, 41), Mod(0, 41), Mod(0, 41), Mod(2, 41), Mod(5, 41), Mod(0, 41), Mod(4, 41), Mod(20, 41), Mod(37, 41), Mod(27, 41), Mod(26, 41), Mod(4, 41), Mod(11, 41), Vecsmall([3]), [41, [9, 31, [6, 0, 0, 0]]], [0, 0, 0, 0]]
EllipticCurve(GF(41), [2,5]).pari_curve() # needs sage.libs.pari
Over a \(p\)-adic field:
sage: # needs sage.libs.pari sage.rings.padics sage: Qp = pAdicField(5, prec=3) sage: E = EllipticCurve(Qp, [3, 4]) sage: E.pari_curve() [0, 0, 0, 3, 4, 0, 6, 16, -9, -144, -3456, -8640, 1728/5, Vecsmall([2]), [O(5^3)], [0, 0]] sage: E.j_invariant() 3*5^-1 + O(5)
>>> from sage.all import * >>> # needs sage.libs.pari sage.rings.padics >>> Qp = pAdicField(Integer(5), prec=Integer(3)) >>> E = EllipticCurve(Qp, [Integer(3), Integer(4)]) >>> E.pari_curve() [0, 0, 0, 3, 4, 0, 6, 16, -9, -144, -3456, -8640, 1728/5, Vecsmall([2]), [O(5^3)], [0, 0]] >>> E.j_invariant() 3*5^-1 + O(5)
# needs sage.libs.pari sage.rings.padics Qp = pAdicField(5, prec=3) E = EllipticCurve(Qp, [3, 4]) E.pari_curve() E.j_invariant()
Over a number field:
sage: K.<a> = QuadraticField(2) # needs sage.libs.pari sage.rings.number_field sage: E = EllipticCurve([1,a]) # needs sage.libs.pari sage.rings.number_field sage: E.pari_curve() # needs sage.libs.pari sage.rings.number_field [0, 0, 0, Mod(1, y^2 - 2), Mod(y, y^2 - 2), 0, Mod(2, y^2 - 2), Mod(4*y, y^2 - 2), Mod(-1, y^2 - 2), Mod(-48, y^2 - 2), Mod(-864*y, y^2 - 2), Mod(-928, y^2 - 2), Mod(3456/29, y^2 - 2), Vecsmall([5]), [[y^2 - 2, [2, 0], 8, 1, [[1, -1.41421356237310; 1, 1.41421356237310], [1, -1.41421356237310; 1, 1.41421356237310], [16, -23; 16, 23], [2, 0; 0, 4], [4, 0; 0, 2], [2, 0; 0, 1], [2, [0, 2; 1, 0]], [2]], [-1.41421356237310, 1.41421356237310], [1, y], [1, 0; 0, 1], [1, 0, 0, 2; 0, 1, 1, 0]]], [0, 0, 0, 0, 0]]
>>> from sage.all import * >>> K = QuadraticField(Integer(2), names=('a',)); (a,) = K._first_ngens(1)# needs sage.libs.pari sage.rings.number_field >>> E = EllipticCurve([Integer(1),a]) # needs sage.libs.pari sage.rings.number_field >>> E.pari_curve() # needs sage.libs.pari sage.rings.number_field [0, 0, 0, Mod(1, y^2 - 2), Mod(y, y^2 - 2), 0, Mod(2, y^2 - 2), Mod(4*y, y^2 - 2), Mod(-1, y^2 - 2), Mod(-48, y^2 - 2), Mod(-864*y, y^2 - 2), Mod(-928, y^2 - 2), Mod(3456/29, y^2 - 2), Vecsmall([5]), [[y^2 - 2, [2, 0], 8, 1, [[1, -1.41421356237310; 1, 1.41421356237310], [1, -1.41421356237310; 1, 1.41421356237310], [16, -23; 16, 23], [2, 0; 0, 4], [4, 0; 0, 2], [2, 0; 0, 1], [2, [0, 2; 1, 0]], [2]], [-1.41421356237310, 1.41421356237310], [1, y], [1, 0; 0, 1], [1, 0, 0, 2; 0, 1, 1, 0]]], [0, 0, 0, 0, 0]]
K.<a> = QuadraticField(2) # needs sage.libs.pari sage.rings.number_field E = EllipticCurve([1,a]) # needs sage.libs.pari sage.rings.number_field E.pari_curve() # needs sage.libs.pari sage.rings.number_field
PARI no longer requires that the \(j\)-invariant has negative \(p\)-adic valuation:
sage: E = EllipticCurve(Qp,[1, 1]) # needs sage.libs.pari sage.rings.padics sage: E.j_invariant() # the j-invariant is a p-adic integer # needs sage.libs.pari sage.rings.padics 2 + 4*5^2 + O(5^3) sage: E.pari_curve() # needs sage.libs.pari sage.rings.padics [0, 0, 0, 1, 1, 0, 2, 4, -1, -48, -864, -496, 6912/31, Vecsmall([2]), [O(5^3)], [0, 0]]
>>> from sage.all import * >>> E = EllipticCurve(Qp,[Integer(1), Integer(1)]) # needs sage.libs.pari sage.rings.padics >>> E.j_invariant() # the j-invariant is a p-adic integer # needs sage.libs.pari sage.rings.padics 2 + 4*5^2 + O(5^3) >>> E.pari_curve() # needs sage.libs.pari sage.rings.padics [0, 0, 0, 1, 1, 0, 2, 4, -1, -48, -864, -496, 6912/31, Vecsmall([2]), [O(5^3)], [0, 0]]
E = EllipticCurve(Qp,[1, 1]) # needs sage.libs.pari sage.rings.padics E.j_invariant() # the j-invariant is a p-adic integer # needs sage.libs.pari sage.rings.padics E.pari_curve() # needs sage.libs.pari sage.rings.padics
- plot(xmin=None, xmax=None, components='both', **args)[source]¶
Draw a graph of this elliptic curve.
The plot method is only implemented when there is a natural coercion from the base ring of
selftoRR. In this case,selfis plotted as if it was defined overRR.INPUT:
xmin,xmax– (optional) points will be computed at least within this range, but possibly farthercomponents– string; one of the following:both– (default), scale so that both bounded and unbounded components appearbounded– scale the plot to show the bounded component. Raises an error if there is only one real component.unbounded– scale the plot to show the unbounded component, including the two flex points
plot_points– passed tosage.plot.generate_plot_points()adaptive_tolerance– passed tosage.plot.generate_plot_points()adaptive_recursion– passed tosage.plot.generate_plot_points()randomize– passed tosage.plot.generate_plot_points()**args– all other options are passed tosage.plot.line.Line
EXAMPLES:
sage: E = EllipticCurve([0, -1]) sage: plot(E, rgbcolor=hue(0.7)) # needs sage.plot Graphics object consisting of 1 graphics primitive sage: E = EllipticCurve('37a') sage: plot(E) # needs sage.plot Graphics object consisting of 2 graphics primitives sage: plot(E, xmin=25, xmax=26) # needs sage.plot Graphics object consisting of 2 graphics primitives
>>> from sage.all import * >>> E = EllipticCurve([Integer(0), -Integer(1)]) >>> plot(E, rgbcolor=hue(RealNumber('0.7'))) # needs sage.plot Graphics object consisting of 1 graphics primitive >>> E = EllipticCurve('37a') >>> plot(E) # needs sage.plot Graphics object consisting of 2 graphics primitives >>> plot(E, xmin=Integer(25), xmax=Integer(26)) # needs sage.plot Graphics object consisting of 2 graphics primitives
E = EllipticCurve([0, -1]) plot(E, rgbcolor=hue(0.7)) # needs sage.plot E = EllipticCurve('37a') plot(E) # needs sage.plot plot(E, xmin=25, xmax=26) # needs sage.plotWith Issue #12766 we added the components keyword:
sage: E.real_components() 2 sage: E.plot(components='bounded') # needs sage.plot Graphics object consisting of 1 graphics primitive sage: E.plot(components='unbounded') # needs sage.plot Graphics object consisting of 1 graphics primitive
>>> from sage.all import * >>> E.real_components() 2 >>> E.plot(components='bounded') # needs sage.plot Graphics object consisting of 1 graphics primitive >>> E.plot(components='unbounded') # needs sage.plot Graphics object consisting of 1 graphics primitive
E.real_components() E.plot(components='bounded') # needs sage.plot E.plot(components='unbounded') # needs sage.plot
If there is only one component then specifying components=’bounded’ raises a ValueError:
sage: E = EllipticCurve('9990be2') sage: E.plot(components='bounded') # needs sage.plot Traceback (most recent call last): ... ValueError: no bounded component for this curve
>>> from sage.all import * >>> E = EllipticCurve('9990be2') >>> E.plot(components='bounded') # needs sage.plot Traceback (most recent call last): ... ValueError: no bounded component for this curve
E = EllipticCurve('9990be2') E.plot(components='bounded') # needs sage.plotAn elliptic curve defined over the Complex Field can not be plotted:
sage: E = EllipticCurve(CC, [0,0,1,-1,0]) sage: E.plot() # needs sage.plot Traceback (most recent call last): ... NotImplementedError: plotting of curves over Complex Field with 53 bits of precision is not implemented yet
>>> from sage.all import * >>> E = EllipticCurve(CC, [Integer(0),Integer(0),Integer(1),-Integer(1),Integer(0)]) >>> E.plot() # needs sage.plot Traceback (most recent call last): ... NotImplementedError: plotting of curves over Complex Field with 53 bits of precision is not implemented yet
E = EllipticCurve(CC, [0,0,1,-1,0]) E.plot() # needs sage.plot
- rst_transform(r, s, t)[source]¶
Return the transform of the curve by \((r,s,t)\) (with \(u=1\)).
INPUT:
r,s,t– three elements of the base ring
OUTPUT:
The elliptic curve obtained from
selfby the standard Weierstrass transformation \((u,r,s,t)\) with \(u=1\).Note
This is just a special case of
change_weierstrass_model(), with \(u=1\).EXAMPLES:
sage: R.<r,s,t> = QQ[] sage: E = EllipticCurve([1,2,3,4,5]) sage: E.rst_transform(r, s, t) Elliptic Curve defined by y^2 + (2*s+1)*x*y + (r+2*t+3)*y = x^3 + (-s^2+3*r-s+2)*x^2 + (3*r^2-r*s-2*s*t+4*r-3*s-t+4)*x + (r^3+2*r^2-r*t-t^2+4*r-3*t+5) over Multivariate Polynomial Ring in r, s, t over Rational Field
>>> from sage.all import * >>> R = QQ['r, s, t']; (r, s, t,) = R._first_ngens(3) >>> E = EllipticCurve([Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)]) >>> E.rst_transform(r, s, t) Elliptic Curve defined by y^2 + (2*s+1)*x*y + (r+2*t+3)*y = x^3 + (-s^2+3*r-s+2)*x^2 + (3*r^2-r*s-2*s*t+4*r-3*s-t+4)*x + (r^3+2*r^2-r*t-t^2+4*r-3*t+5) over Multivariate Polynomial Ring in r, s, t over Rational Field
R.<r,s,t> = QQ[] E = EllipticCurve([1,2,3,4,5]) E.rst_transform(r, s, t)
- scalar_multiplication(m)[source]¶
Return the scalar-multiplication map \([m]\) on this elliptic curve as a
sage.schemes.elliptic_curves.hom_scalar.EllipticCurveHom_scalarobject.EXAMPLES:
sage: E = EllipticCurve('77a1') sage: m = E.scalar_multiplication(-7); m Scalar-multiplication endomorphism [-7] of Elliptic Curve defined by y^2 + y = x^3 + 2*x over Rational Field sage: m.degree() 49 sage: P = E(2,3) sage: m(P) (-26/225 : -2132/3375 : 1) sage: m.rational_maps() == E.multiplication_by_m(-7) True
>>> from sage.all import * >>> E = EllipticCurve('77a1') >>> m = E.scalar_multiplication(-Integer(7)); m Scalar-multiplication endomorphism [-7] of Elliptic Curve defined by y^2 + y = x^3 + 2*x over Rational Field >>> m.degree() 49 >>> P = E(Integer(2),Integer(3)) >>> m(P) (-26/225 : -2132/3375 : 1) >>> m.rational_maps() == E.multiplication_by_m(-Integer(7)) True
E = EllipticCurve('77a1') m = E.scalar_multiplication(-7); m m.degree() P = E(2,3) m(P) m.rational_maps() == E.multiplication_by_m(-7)sage: E = EllipticCurve('11a1') sage: E.scalar_multiplication(7) Scalar-multiplication endomorphism [7] of Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field
>>> from sage.all import * >>> E = EllipticCurve('11a1') >>> E.scalar_multiplication(Integer(7)) Scalar-multiplication endomorphism [7] of Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field
E = EllipticCurve('11a1') E.scalar_multiplication(7)>>> from sage.all import * >>> E = EllipticCurve('11a1') >>> E.scalar_multiplication(Integer(7)) Scalar-multiplication endomorphism [7] of Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field
E = EllipticCurve('11a1') E.scalar_multiplication(7)
- scale_curve(u)[source]¶
Return the transform of the curve by scale factor \(u\).
INPUT:
u– an invertible element of the base ring
OUTPUT:
The elliptic curve obtained from
selfby the standard Weierstrass transformation \((u,r,s,t)\) with \(r=s=t=0\).Note
This is just a special case of
change_weierstrass_model(), with \(r=s=t=0\).EXAMPLES:
sage: K = Frac(PolynomialRing(QQ, 'u')) sage: u = K.gen() sage: E = EllipticCurve([1,2,3,4,5]) sage: E.scale_curve(u) Elliptic Curve defined by y^2 + u*x*y + 3*u^3*y = x^3 + 2*u^2*x^2 + 4*u^4*x + 5*u^6 over Fraction Field of Univariate Polynomial Ring in u over Rational Field
>>> from sage.all import * >>> K = Frac(PolynomialRing(QQ, 'u')) >>> u = K.gen() >>> E = EllipticCurve([Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)]) >>> E.scale_curve(u) Elliptic Curve defined by y^2 + u*x*y + 3*u^3*y = x^3 + 2*u^2*x^2 + 4*u^4*x + 5*u^6 over Fraction Field of Univariate Polynomial Ring in u over Rational Field
K = Frac(PolynomialRing(QQ, 'u')) u = K.gen() E = EllipticCurve([1,2,3,4,5]) E.scale_curve(u)
- short_weierstrass_model(complete_cube=True)[source]¶
Return a short Weierstrass model for
self.INPUT:
complete_cube– boolean (default:True); for meaning, see below
OUTPUT: an elliptic curve
If
complete_cube=True: Return a model of the form \(y^2 = x^3 + a*x + b\) for this curve. The characteristic must not be 2; in characteristic 3, it is only possible if \(b_2=0\).If
complete_cube=False: Return a model of the form \(y^2 = x^3 + ax^2 + bx + c\) for this curve. The characteristic must not be 2.EXAMPLES:
sage: E = EllipticCurve([1,2,3,4,5]) sage: E Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Rational Field sage: F = E.short_weierstrass_model() sage: F Elliptic Curve defined by y^2 = x^3 + 4941*x + 185166 over Rational Field sage: E.is_isomorphic(F) True sage: F = E.short_weierstrass_model(complete_cube=False) sage: F Elliptic Curve defined by y^2 = x^3 + 9*x^2 + 88*x + 464 over Rational Field sage: E.is_isomorphic(F) True
>>> from sage.all import * >>> E = EllipticCurve([Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)]) >>> E Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Rational Field >>> F = E.short_weierstrass_model() >>> F Elliptic Curve defined by y^2 = x^3 + 4941*x + 185166 over Rational Field >>> E.is_isomorphic(F) True >>> F = E.short_weierstrass_model(complete_cube=False) >>> F Elliptic Curve defined by y^2 = x^3 + 9*x^2 + 88*x + 464 over Rational Field >>> E.is_isomorphic(F) True
E = EllipticCurve([1,2,3,4,5]) E F = E.short_weierstrass_model() F E.is_isomorphic(F) F = E.short_weierstrass_model(complete_cube=False) F E.is_isomorphic(F)
sage: E = EllipticCurve(GF(3), [1,2,3,4,5]) sage: E.short_weierstrass_model(complete_cube=False) Elliptic Curve defined by y^2 = x^3 + x + 2 over Finite Field of size 3
>>> from sage.all import * >>> E = EllipticCurve(GF(Integer(3)), [Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)]) >>> E.short_weierstrass_model(complete_cube=False) Elliptic Curve defined by y^2 = x^3 + x + 2 over Finite Field of size 3
E = EllipticCurve(GF(3), [1,2,3,4,5]) E.short_weierstrass_model(complete_cube=False)
>>> from sage.all import * >>> E = EllipticCurve(GF(Integer(3)), [Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)]) >>> E.short_weierstrass_model(complete_cube=False) Elliptic Curve defined by y^2 = x^3 + x + 2 over Finite Field of size 3
E = EllipticCurve(GF(3), [1,2,3,4,5]) E.short_weierstrass_model(complete_cube=False)
This used to be different see Issue #3973:
sage: E.short_weierstrass_model() Elliptic Curve defined by y^2 = x^3 + x + 2 over Finite Field of size 3
>>> from sage.all import * >>> E.short_weierstrass_model() Elliptic Curve defined by y^2 = x^3 + x + 2 over Finite Field of size 3
E.short_weierstrass_model()
More tests in characteristic 3:
sage: E = EllipticCurve(GF(3), [0,2,1,2,1]) sage: E.short_weierstrass_model() Traceback (most recent call last): ... ValueError: short_weierstrass_model(): no short model for Elliptic Curve defined by y^2 + y = x^3 + 2*x^2 + 2*x + 1 over Finite Field of size 3 (characteristic is 3) sage: E.short_weierstrass_model(complete_cube=False) Elliptic Curve defined by y^2 = x^3 + 2*x^2 + 2*x + 2 over Finite Field of size 3 sage: E.short_weierstrass_model(complete_cube=False).is_isomorphic(E) True
>>> from sage.all import * >>> E = EllipticCurve(GF(Integer(3)), [Integer(0),Integer(2),Integer(1),Integer(2),Integer(1)]) >>> E.short_weierstrass_model() Traceback (most recent call last): ... ValueError: short_weierstrass_model(): no short model for Elliptic Curve defined by y^2 + y = x^3 + 2*x^2 + 2*x + 1 over Finite Field of size 3 (characteristic is 3) >>> E.short_weierstrass_model(complete_cube=False) Elliptic Curve defined by y^2 = x^3 + 2*x^2 + 2*x + 2 over Finite Field of size 3 >>> E.short_weierstrass_model(complete_cube=False).is_isomorphic(E) True
E = EllipticCurve(GF(3), [0,2,1,2,1]) E.short_weierstrass_model() E.short_weierstrass_model(complete_cube=False) E.short_weierstrass_model(complete_cube=False).is_isomorphic(E)
- torsion_polynomial(m, x=None, two_torsion_multiplicity=2, force_evaluate=None)[source]¶
Return the \(m\)-th division polynomial of this elliptic curve evaluated at \(x\).
The division polynomial is cached if \(x\) is
None.INPUT:
m– positive integerx– (optional) ring element to use as the \(x\) variable. If \(x\) isNone(omitted), then a new polynomial ring will be constructed over the base ring of the elliptic curve, and its generator will be used as \(x\). Note that \(x\) does not need to be a generator of a polynomial ring; any ring element works. This permits fast calculation of the torsion polynomial evaluated on any element of a ring.two_torsion_multiplicity– 0, 1, or 2If 0: For even \(m\) when \(x\) is
None, a univariate polynomial over the base ring of the curve is returned, which omits factors whose roots are the \(x\)-coordinates of the \(2\)-torsion points. When \(x\) is notNone, the evaluation of such a polynomial at \(x\) is returned.If 2: For even \(m\) when \(x\) is
None, a univariate polynomial over the base ring of the curve is returned, which includes a factor of degree 3 whose roots are the \(x\)-coordinates of the \(2\)-torsion points. Similarly, when \(x\) is notNone, the evaluation of such a polynomial at \(x\) is returned.If 1: For even \(m\) when \(x\) is
None, a bivariate polynomial over the base ring of the curve is returned, which includes a factor \(2y+a_1x+a_3\) having simple zeros at the \(2\)-torsion points. When \(x\) is notNone, it should be a tuple of length 2, and the evaluation of such a polynomial at \(x\) is returned.force_evaluate– (optional) 0, 1, or 2By default, this method makes use of previously cached generic division polynomials to compute the value of the polynomial at a given element \(x\) whenever it appears beneficial to do so. Explicitly setting this flag overrides the default behavior.
Note that the complexity of evaluating a generic division polynomial scales much worse than that of computing the value at a point directly (using the recursive formulas), hence setting this flag can be detrimental to performance.
If 0: Do not use cached generic division polynomials.
If 1: If the generic division polynomial for this \(m\) has been cached before, evaluate it at \(x\) to compute the result.
If 2: Compute the value at \(x\) by evaluating the generic division polynomial. If the generic \(m\)-division polynomial has not yet been cached, compute and cache it first.
EXAMPLES:
sage: E = EllipticCurve([0,0,1,-1,0]) sage: E.division_polynomial(1) 1 sage: E.division_polynomial(2, two_torsion_multiplicity=0) 1 sage: E.division_polynomial(2, two_torsion_multiplicity=1) 2*y + 1 sage: E.division_polynomial(2, two_torsion_multiplicity=2) 4*x^3 - 4*x + 1 sage: E.division_polynomial(2) 4*x^3 - 4*x + 1 sage: [E.division_polynomial(3, two_torsion_multiplicity=i) for i in range(3)] [3*x^4 - 6*x^2 + 3*x - 1, 3*x^4 - 6*x^2 + 3*x - 1, 3*x^4 - 6*x^2 + 3*x - 1] sage: [type(E.division_polynomial(3, two_torsion_multiplicity=i)) for i in range(3)] [<... 'sage.rings.polynomial.polynomial_rational_flint.Polynomial_rational_flint'>, <... 'sage.rings.polynomial.multi_polynomial_libsingular.MPolynomial_libsingular'>, <... 'sage.rings.polynomial.polynomial_rational_flint.Polynomial_rational_flint'>]
>>> from sage.all import * >>> E = EllipticCurve([Integer(0),Integer(0),Integer(1),-Integer(1),Integer(0)]) >>> E.division_polynomial(Integer(1)) 1 >>> E.division_polynomial(Integer(2), two_torsion_multiplicity=Integer(0)) 1 >>> E.division_polynomial(Integer(2), two_torsion_multiplicity=Integer(1)) 2*y + 1 >>> E.division_polynomial(Integer(2), two_torsion_multiplicity=Integer(2)) 4*x^3 - 4*x + 1 >>> E.division_polynomial(Integer(2)) 4*x^3 - 4*x + 1 >>> [E.division_polynomial(Integer(3), two_torsion_multiplicity=i) for i in range(Integer(3))] [3*x^4 - 6*x^2 + 3*x - 1, 3*x^4 - 6*x^2 + 3*x - 1, 3*x^4 - 6*x^2 + 3*x - 1] >>> [type(E.division_polynomial(Integer(3), two_torsion_multiplicity=i)) for i in range(Integer(3))] [<... 'sage.rings.polynomial.polynomial_rational_flint.Polynomial_rational_flint'>, <... 'sage.rings.polynomial.multi_polynomial_libsingular.MPolynomial_libsingular'>, <... 'sage.rings.polynomial.polynomial_rational_flint.Polynomial_rational_flint'>]
E = EllipticCurve([0,0,1,-1,0]) E.division_polynomial(1) E.division_polynomial(2, two_torsion_multiplicity=0) E.division_polynomial(2, two_torsion_multiplicity=1) E.division_polynomial(2, two_torsion_multiplicity=2) E.division_polynomial(2) [E.division_polynomial(3, two_torsion_multiplicity=i) for i in range(3)] [type(E.division_polynomial(3, two_torsion_multiplicity=i)) for i in range(3)]
sage: E = EllipticCurve([0, -1, 1, -10, -20]) sage: R.<z> = PolynomialRing(QQ) sage: E.division_polynomial(4, z, 0) 2*z^6 - 4*z^5 - 100*z^4 - 790*z^3 - 210*z^2 - 1496*z - 5821 sage: E.division_polynomial(4, z) 8*z^9 - 24*z^8 - 464*z^7 - 2758*z^6 + 6636*z^5 + 34356*z^4 + 53510*z^3 + 99714*z^2 + 351024*z + 459859
>>> from sage.all import * >>> E = EllipticCurve([Integer(0), -Integer(1), Integer(1), -Integer(10), -Integer(20)]) >>> R = PolynomialRing(QQ, names=('z',)); (z,) = R._first_ngens(1) >>> E.division_polynomial(Integer(4), z, Integer(0)) 2*z^6 - 4*z^5 - 100*z^4 - 790*z^3 - 210*z^2 - 1496*z - 5821 >>> E.division_polynomial(Integer(4), z) 8*z^9 - 24*z^8 - 464*z^7 - 2758*z^6 + 6636*z^5 + 34356*z^4 + 53510*z^3 + 99714*z^2 + 351024*z + 459859
E = EllipticCurve([0, -1, 1, -10, -20]) R.<z> = PolynomialRing(QQ) E.division_polynomial(4, z, 0) E.division_polynomial(4, z)
>>> from sage.all import * >>> E = EllipticCurve([Integer(0), -Integer(1), Integer(1), -Integer(10), -Integer(20)]) >>> R = PolynomialRing(QQ, names=('z',)); (z,) = R._first_ngens(1) >>> E.division_polynomial(Integer(4), z, Integer(0)) 2*z^6 - 4*z^5 - 100*z^4 - 790*z^3 - 210*z^2 - 1496*z - 5821 >>> E.division_polynomial(Integer(4), z) 8*z^9 - 24*z^8 - 464*z^7 - 2758*z^6 + 6636*z^5 + 34356*z^4 + 53510*z^3 + 99714*z^2 + 351024*z + 459859
E = EllipticCurve([0, -1, 1, -10, -20]) R.<z> = PolynomialRing(QQ) E.division_polynomial(4, z, 0) E.division_polynomial(4, z)
This does not work, since when two_torsion_multiplicity is 1, we compute a bivariate polynomial, and must evaluate at a tuple of length 2:
sage: E.division_polynomial(4,z,1) Traceback (most recent call last): ... ValueError: x should be a tuple of length 2 (or None) when two_torsion_multiplicity is 1 sage: R.<z,w> = PolynomialRing(QQ, 2) sage: E.division_polynomial(4, (z,w), 1).factor() (2*w + 1) * (2*z^6 - 4*z^5 - 100*z^4 - 790*z^3 - 210*z^2 - 1496*z - 5821)
>>> from sage.all import * >>> E.division_polynomial(Integer(4),z,Integer(1)) Traceback (most recent call last): ... ValueError: x should be a tuple of length 2 (or None) when two_torsion_multiplicity is 1 >>> R = PolynomialRing(QQ, Integer(2), names=('z', 'w',)); (z, w,) = R._first_ngens(2) >>> E.division_polynomial(Integer(4), (z,w), Integer(1)).factor() (2*w + 1) * (2*z^6 - 4*z^5 - 100*z^4 - 790*z^3 - 210*z^2 - 1496*z - 5821)
E.division_polynomial(4,z,1) R.<z,w> = PolynomialRing(QQ, 2) E.division_polynomial(4, (z,w), 1).factor()
We can also evaluate this bivariate polynomial at a point:
sage: P = E(5,5) sage: E.division_polynomial(4,P,two_torsion_multiplicity=1) -1771561
>>> from sage.all import * >>> P = E(Integer(5),Integer(5)) >>> E.division_polynomial(Integer(4),P,two_torsion_multiplicity=Integer(1)) -1771561
P = E(5,5) E.division_polynomial(4,P,two_torsion_multiplicity=1)
- two_division_polynomial(x=None)[source]¶
Return the 2-division polynomial of this elliptic curve evaluated at
x.INPUT:
x– (optional) ring element to use as the \(x\) variable. IfxisNone, then a new polynomial ring will be constructed over the base ring of the elliptic curve, and its generator will be used asx. Note thatxdoes not need to be a generator of a polynomial ring; any ring element is acceptable. This permits fast calculation of the torsion polynomial evaluated on any element of a ring.
EXAMPLES:
sage: E = EllipticCurve('5077a1') sage: E.two_division_polynomial() 4*x^3 - 28*x + 25 sage: E = EllipticCurve(GF(3^2,'a'), [1,1,1,1,1]) # needs sage.rings.finite_rings sage: E.two_division_polynomial() # needs sage.rings.finite_rings x^3 + 2*x^2 + 2 sage: E.two_division_polynomial().roots() # needs sage.rings.finite_rings [(2, 1), (2*a, 1), (a + 2, 1)]
>>> from sage.all import * >>> E = EllipticCurve('5077a1') >>> E.two_division_polynomial() 4*x^3 - 28*x + 25 >>> E = EllipticCurve(GF(Integer(3)**Integer(2),'a'), [Integer(1),Integer(1),Integer(1),Integer(1),Integer(1)]) # needs sage.rings.finite_rings >>> E.two_division_polynomial() # needs sage.rings.finite_rings x^3 + 2*x^2 + 2 >>> E.two_division_polynomial().roots() # needs sage.rings.finite_rings [(2, 1), (2*a, 1), (a + 2, 1)]
E = EllipticCurve('5077a1') E.two_division_polynomial() E = EllipticCurve(GF(3^2,'a'), [1,1,1,1,1]) # needs sage.rings.finite_rings E.two_division_polynomial() # needs sage.rings.finite_rings E.two_division_polynomial().roots() # needs sage.rings.finite_rings
- sage.schemes.elliptic_curves.ell_generic.is_EllipticCurve(x)[source]¶
Utility function to test if
xis an instance of an Elliptic Curve class.EXAMPLES:
sage: from sage.schemes.elliptic_curves.ell_generic import is_EllipticCurve sage: E = EllipticCurve([1,2,3/4,7,19]) sage: is_EllipticCurve(E) doctest:warning... DeprecationWarning: The function is_EllipticCurve is deprecated; use 'isinstance(..., EllipticCurve_generic)' instead. See https://github.com/sagemath/sage/issues/38022 for details. True sage: is_EllipticCurve(0) False
>>> from sage.all import * >>> from sage.schemes.elliptic_curves.ell_generic import is_EllipticCurve >>> E = EllipticCurve([Integer(1),Integer(2),Integer(3)/Integer(4),Integer(7),Integer(19)]) >>> is_EllipticCurve(E) doctest:warning... DeprecationWarning: The function is_EllipticCurve is deprecated; use 'isinstance(..., EllipticCurve_generic)' instead. See https://github.com/sagemath/sage/issues/38022 for details. True >>> is_EllipticCurve(Integer(0)) False
from sage.schemes.elliptic_curves.ell_generic import is_EllipticCurve E = EllipticCurve([1,2,3/4,7,19]) is_EllipticCurve(E) is_EllipticCurve(0)