Some More Advanced Mathematics¶
Algebraic Geometry¶
You can define arbitrary algebraic varieties in Sage, but sometimes nontrivial functionality is limited to rings over \(\QQ\) or finite fields. For example, we compute the union of two affine plane curves, then recover the curves as the irreducible components of the union.
sage: x, y = AffineSpace(2, QQ, 'xy').gens()
sage: C2 = Curve(x^2 + y^2 - 1)
sage: C3 = Curve(x^3 + y^3 - 1)
sage: D = C2 + C3
sage: D
Affine Plane Curve over Rational Field defined by
x^5 + x^3*y^2 + x^2*y^3 + y^5 - x^3 - y^3 - x^2 - y^2 + 1
sage: D.irreducible_components()
[
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
x^2 + y^2 - 1,
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
x^3 + y^3 - 1
]
>>> from sage.all import *
>>> x, y = AffineSpace(Integer(2), QQ, 'xy').gens()
>>> C2 = Curve(x**Integer(2) + y**Integer(2) - Integer(1))
>>> C3 = Curve(x**Integer(3) + y**Integer(3) - Integer(1))
>>> D = C2 + C3
>>> D
Affine Plane Curve over Rational Field defined by
x^5 + x^3*y^2 + x^2*y^3 + y^5 - x^3 - y^3 - x^2 - y^2 + 1
>>> D.irreducible_components()
[
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
x^2 + y^2 - 1,
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
x^3 + y^3 - 1
]
x, y = AffineSpace(2, QQ, 'xy').gens() C2 = Curve(x^2 + y^2 - 1) C3 = Curve(x^3 + y^3 - 1) D = C2 + C3 D D.irreducible_components()
We can also find all points of intersection of the two curves by intersecting them and computing the irreducible components.
sage: V = C2.intersection(C3)
sage: V.irreducible_components()
[
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
y,
x - 1,
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
y - 1,
x,
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
x + y + 2,
2*y^2 + 4*y + 3
]
>>> from sage.all import *
>>> V = C2.intersection(C3)
>>> V.irreducible_components()
[
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
y,
x - 1,
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
y - 1,
x,
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
x + y + 2,
2*y^2 + 4*y + 3
]
V = C2.intersection(C3) V.irreducible_components()
Thus, e.g., \((1,0)\) and \((0,1)\) are on both curves (visibly clear), as are certain (quadratic) points whose \(y\) coordinates satisfy \(2y^2 + 4y + 3=0\).
Sage can compute the toric ideal of the twisted cubic in projective 3 space:
sage: R.<a,b,c,d> = PolynomialRing(QQ, 4)
sage: I = ideal(b^2-a*c, c^2-b*d, a*d-b*c)
sage: F = I.groebner_fan(); F
Groebner fan of the ideal:
Ideal (b^2 - a*c, c^2 - b*d, -b*c + a*d) of Multivariate Polynomial Ring
in a, b, c, d over Rational Field
sage: F.reduced_groebner_bases ()
[[-c^2 + b*d, -b*c + a*d, -b^2 + a*c],
[-b*c + a*d, -c^2 + b*d, b^2 - a*c],
[-c^3 + a*d^2, -c^2 + b*d, b*c - a*d, b^2 - a*c],
[-c^2 + b*d, b^2 - a*c, b*c - a*d, c^3 - a*d^2],
[-b*c + a*d, -b^2 + a*c, c^2 - b*d],
[-b^3 + a^2*d, -b^2 + a*c, c^2 - b*d, b*c - a*d],
[-b^2 + a*c, c^2 - b*d, b*c - a*d, b^3 - a^2*d],
[c^2 - b*d, b*c - a*d, b^2 - a*c]]
sage: F.polyhedralfan()
Polyhedral fan in 4 dimensions of dimension 4
>>> from sage.all import *
>>> R = PolynomialRing(QQ, Integer(4), names=('a', 'b', 'c', 'd',)); (a, b, c, d,) = R._first_ngens(4)
>>> I = ideal(b**Integer(2)-a*c, c**Integer(2)-b*d, a*d-b*c)
>>> F = I.groebner_fan(); F
Groebner fan of the ideal:
Ideal (b^2 - a*c, c^2 - b*d, -b*c + a*d) of Multivariate Polynomial Ring
in a, b, c, d over Rational Field
>>> F.reduced_groebner_bases ()
[[-c^2 + b*d, -b*c + a*d, -b^2 + a*c],
[-b*c + a*d, -c^2 + b*d, b^2 - a*c],
[-c^3 + a*d^2, -c^2 + b*d, b*c - a*d, b^2 - a*c],
[-c^2 + b*d, b^2 - a*c, b*c - a*d, c^3 - a*d^2],
[-b*c + a*d, -b^2 + a*c, c^2 - b*d],
[-b^3 + a^2*d, -b^2 + a*c, c^2 - b*d, b*c - a*d],
[-b^2 + a*c, c^2 - b*d, b*c - a*d, b^3 - a^2*d],
[c^2 - b*d, b*c - a*d, b^2 - a*c]]
>>> F.polyhedralfan()
Polyhedral fan in 4 dimensions of dimension 4
R.<a,b,c,d> = PolynomialRing(QQ, 4) I = ideal(b^2-a*c, c^2-b*d, a*d-b*c) F = I.groebner_fan(); F F.reduced_groebner_bases () F.polyhedralfan()
Elliptic Curves¶
Elliptic curve functionality includes most of the elliptic curve functionality of PARI, access to the data in Cremona’s online tables (this requires an optional database package), the functionality of mwrank, i.e., 2-descents with computation of the full Mordell-Weil group, the SEA algorithm, computation of all isogenies, much new code for curves over \(\QQ\), and some of Denis Simon’s algebraic descent software.
The command EllipticCurve for creating an elliptic curve has many
forms:
EllipticCurve([\(a_1\), \(a_2\), \(a_3\), \(a_4\), \(a_6\)]): Returns the elliptic curve
\[y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6,\]where the \(a_i\)’s are coerced into the parent of \(a_1\). If all the \(a_i\) have parent \(\ZZ\), they are coerced into \(\QQ\).
EllipticCurve([\(a_4\), \(a_6\)]): Same as above, but \(a_1=a_2=a_3=0\).
EllipticCurve(label): Returns the elliptic curve over from the Cremona database with the given (new!) Cremona label. The label is a string, such as
"11a"or"37b2". The letter must be lower case (to distinguish it from the old labeling).EllipticCurve(j): Returns an elliptic curve with \(j\)-invariant \(j\).
EllipticCurve(R, [\(a_1\), \(a_2\), \(a_3\), \(a_4\), \(a_6\)]): Create the elliptic curve over a ring \(R\) with given \(a_i\)’s as above.
We illustrate each of these constructors:
sage: EllipticCurve([0,0,1,-1,0])
Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
sage: EllipticCurve([GF(5)(0),0,1,-1,0])
Elliptic Curve defined by y^2 + y = x^3 + 4*x over Finite Field of size 5
sage: EllipticCurve([1,2])
Elliptic Curve defined by y^2 = x^3 + x + 2 over Rational Field
sage: EllipticCurve('37a')
Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
sage: EllipticCurve_from_j(1)
Elliptic Curve defined by y^2 + x*y = x^3 + 36*x + 3455 over Rational Field
sage: EllipticCurve(GF(5), [0,0,1,-1,0])
Elliptic Curve defined by y^2 + y = x^3 + 4*x over Finite Field of size 5
>>> from sage.all import *
>>> EllipticCurve([Integer(0),Integer(0),Integer(1),-Integer(1),Integer(0)])
Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
>>> EllipticCurve([GF(Integer(5))(Integer(0)),Integer(0),Integer(1),-Integer(1),Integer(0)])
Elliptic Curve defined by y^2 + y = x^3 + 4*x over Finite Field of size 5
>>> EllipticCurve([Integer(1),Integer(2)])
Elliptic Curve defined by y^2 = x^3 + x + 2 over Rational Field
>>> EllipticCurve('37a')
Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
>>> EllipticCurve_from_j(Integer(1))
Elliptic Curve defined by y^2 + x*y = x^3 + 36*x + 3455 over Rational Field
>>> EllipticCurve(GF(Integer(5)), [Integer(0),Integer(0),Integer(1),-Integer(1),Integer(0)])
Elliptic Curve defined by y^2 + y = x^3 + 4*x over Finite Field of size 5
EllipticCurve([0,0,1,-1,0])
EllipticCurve([GF(5)(0),0,1,-1,0])
EllipticCurve([1,2])
EllipticCurve('37a')
EllipticCurve_from_j(1)
EllipticCurve(GF(5), [0,0,1,-1,0])
The pair \((0,0)\) is a point on the elliptic curve
\(E\) defined by \(y^2 +
y = x^3 - x\). To create this
point in Sage type E([0,0]). Sage can add points on such an
elliptic curve (recall elliptic curves support an additive group
structure where the point at infinity is the zero element and three
co-linear points on the curve add to zero):
sage: E = EllipticCurve([0,0,1,-1,0])
sage: E
Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
sage: P = E([0,0])
sage: P + P
(1 : 0 : 1)
sage: 10*P
(161/16 : -2065/64 : 1)
sage: 20*P
(683916417/264517696 : -18784454671297/4302115807744 : 1)
sage: E.conductor()
37
>>> from sage.all import *
>>> E = EllipticCurve([Integer(0),Integer(0),Integer(1),-Integer(1),Integer(0)])
>>> E
Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
>>> P = E([Integer(0),Integer(0)])
>>> P + P
(1 : 0 : 1)
>>> Integer(10)*P
(161/16 : -2065/64 : 1)
>>> Integer(20)*P
(683916417/264517696 : -18784454671297/4302115807744 : 1)
>>> E.conductor()
37
E = EllipticCurve([0,0,1,-1,0]) E P = E([0,0]) P + P 10*P 20*P E.conductor()
The elliptic curves over the complex numbers are parameterized by the \(j\)-invariant. Sage computes \(j\)-invariant as follows:
sage: E = EllipticCurve([0,0,0,-4,2]); E
Elliptic Curve defined by y^2 = x^3 - 4*x + 2 over Rational Field
sage: E.conductor()
2368
sage: E.j_invariant()
110592/37
>>> from sage.all import *
>>> E = EllipticCurve([Integer(0),Integer(0),Integer(0),-Integer(4),Integer(2)]); E
Elliptic Curve defined by y^2 = x^3 - 4*x + 2 over Rational Field
>>> E.conductor()
2368
>>> E.j_invariant()
110592/37
E = EllipticCurve([0,0,0,-4,2]); E E.conductor() E.j_invariant()
If we make a curve with the same \(j\)-invariant as that of \(E\), it need not be isomorphic to \(E\). In the following example, the curves are not isomorphic because their conductors are different.
sage: F = EllipticCurve_from_j(110592/37)
sage: F.conductor()
37
>>> from sage.all import *
>>> F = EllipticCurve_from_j(Integer(110592)/Integer(37))
>>> F.conductor()
37
F = EllipticCurve_from_j(110592/37) F.conductor()
However, the twist of \(F\) by 2 gives an isomorphic curve.
sage: G = F.quadratic_twist(2); G
Elliptic Curve defined by y^2 = x^3 - 4*x + 2 over Rational Field
sage: G.conductor()
2368
sage: G.j_invariant()
110592/37
>>> from sage.all import *
>>> G = F.quadratic_twist(Integer(2)); G
Elliptic Curve defined by y^2 = x^3 - 4*x + 2 over Rational Field
>>> G.conductor()
2368
>>> G.j_invariant()
110592/37
G = F.quadratic_twist(2); G G.conductor() G.j_invariant()
We can compute the coefficients \(a_n\) of the \(L\)-series or modular form \(\sum_{n=0}^\infty a_nq^n\) attached to the elliptic curve. This computation uses the PARI C-library:
sage: E = EllipticCurve([0,0,1,-1,0])
sage: E.anlist(30)
[0, 1, -2, -3, 2, -2, 6, -1, 0, 6, 4, -5, -6, -2, 2, 6, -4, 0, -12, 0, -4,
3, 10, 2, 0, -1, 4, -9, -2, 6, -12]
sage: v = E.anlist(10000)
>>> from sage.all import *
>>> E = EllipticCurve([Integer(0),Integer(0),Integer(1),-Integer(1),Integer(0)])
>>> E.anlist(Integer(30))
[0, 1, -2, -3, 2, -2, 6, -1, 0, 6, 4, -5, -6, -2, 2, 6, -4, 0, -12, 0, -4,
3, 10, 2, 0, -1, 4, -9, -2, 6, -12]
>>> v = E.anlist(Integer(10000))
E = EllipticCurve([0,0,1,-1,0]) E.anlist(30) v = E.anlist(10000)
It only takes a second to compute all \(a_n\) for \(n\leq 10^5\):
sage: %time v = E.anlist(100000)
CPU times: user 0.98 s, sys: 0.06 s, total: 1.04 s
Wall time: 1.06
>>> from sage.all import *
>>> %time v = E.anlist(Integer(100000))
CPU times: user 0.98 s, sys: 0.06 s, total: 1.04 s
Wall time: 1.06
%time v = E.anlist(100000)
Elliptic curves can be constructed using their Cremona labels. This pre-loads the elliptic curve with information about its rank, Tamagawa numbers, regulator, etc.
sage: E = EllipticCurve("37b2")
sage: E
Elliptic Curve defined by y^2 + y = x^3 + x^2 - 1873*x - 31833 over Rational
Field
sage: E = EllipticCurve("389a")
sage: E
Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field
sage: E.rank()
2
sage: E = EllipticCurve("5077a")
sage: E.rank()
3
>>> from sage.all import *
>>> E = EllipticCurve("37b2")
>>> E
Elliptic Curve defined by y^2 + y = x^3 + x^2 - 1873*x - 31833 over Rational
Field
>>> E = EllipticCurve("389a")
>>> E
Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field
>>> E.rank()
2
>>> E = EllipticCurve("5077a")
>>> E.rank()
3
E = EllipticCurve("37b2")
E
E = EllipticCurve("389a")
E
E.rank()
E = EllipticCurve("5077a")
E.rank()
We can also access the Cremona database directly.
sage: db = sage.databases.cremona.CremonaDatabase()
sage: db.curves(37)
{'a1': [[0, 0, 1, -1, 0], 1, 1], 'b1': [[0, 1, 1, -23, -50], 0, 3]}
sage: db.allcurves(37)
{'a1': [[0, 0, 1, -1, 0], 1, 1],
'b1': [[0, 1, 1, -23, -50], 0, 3],
'b2': [[0, 1, 1, -1873, -31833], 0, 1],
'b3': [[0, 1, 1, -3, 1], 0, 3]}
>>> from sage.all import *
>>> db = sage.databases.cremona.CremonaDatabase()
>>> db.curves(Integer(37))
{'a1': [[0, 0, 1, -1, 0], 1, 1], 'b1': [[0, 1, 1, -23, -50], 0, 3]}
>>> db.allcurves(Integer(37))
{'a1': [[0, 0, 1, -1, 0], 1, 1],
'b1': [[0, 1, 1, -23, -50], 0, 3],
'b2': [[0, 1, 1, -1873, -31833], 0, 1],
'b3': [[0, 1, 1, -3, 1], 0, 3]}
db = sage.databases.cremona.CremonaDatabase() db.curves(37) db.allcurves(37)
The objects returned from the database are not of type
EllipticCurve. They are elements of a database and have a couple
of fields, and that’s it. There is a small version of Cremona’s
database, which is distributed by default with Sage, and contains
limited information about elliptic curves of conductor
\(\leq 10000\). There is also a large optional version, which
contains extensive data about all curves of conductor up to
\(120000\) (as of October 2005). There is also a huge (2GB)
optional database package for Sage that contains the hundreds of
millions of elliptic curves in the Stein-Watkins database.
Dirichlet Characters¶
A Dirichlet character is the extension of a homomorphism \((\ZZ/N\ZZ)^* \to R^*\), for some ring \(R\), to the map \(\ZZ \to R\) obtained by sending those integers \(x\) with \(\gcd(N,x)>1\) to 0.
sage: G = DirichletGroup(12)
sage: G.list()
[Dirichlet character modulo 12 of conductor 1 mapping 7 |--> 1, 5 |--> 1,
Dirichlet character modulo 12 of conductor 4 mapping 7 |--> -1, 5 |--> 1,
Dirichlet character modulo 12 of conductor 3 mapping 7 |--> 1, 5 |--> -1,
Dirichlet character modulo 12 of conductor 12 mapping 7 |--> -1, 5 |--> -1]
sage: G.gens()
(Dirichlet character modulo 12 of conductor 4 mapping 7 |--> -1, 5 |--> 1,
Dirichlet character modulo 12 of conductor 3 mapping 7 |--> 1, 5 |--> -1)
sage: len(G)
4
>>> from sage.all import *
>>> G = DirichletGroup(Integer(12))
>>> G.list()
[Dirichlet character modulo 12 of conductor 1 mapping 7 |--> 1, 5 |--> 1,
Dirichlet character modulo 12 of conductor 4 mapping 7 |--> -1, 5 |--> 1,
Dirichlet character modulo 12 of conductor 3 mapping 7 |--> 1, 5 |--> -1,
Dirichlet character modulo 12 of conductor 12 mapping 7 |--> -1, 5 |--> -1]
>>> G.gens()
(Dirichlet character modulo 12 of conductor 4 mapping 7 |--> -1, 5 |--> 1,
Dirichlet character modulo 12 of conductor 3 mapping 7 |--> 1, 5 |--> -1)
>>> len(G)
4
G = DirichletGroup(12) G.list() G.gens() len(G)
Having created the group, we next create an element and compute with it.
sage: G = DirichletGroup(21)
sage: chi = G.1; chi
Dirichlet character modulo 21 of conductor 7 mapping 8 |--> 1, 10 |--> zeta6
sage: chi.values()
[0, 1, zeta6 - 1, 0, -zeta6, -zeta6 + 1, 0, 0, 1, 0, zeta6, -zeta6, 0, -1,
0, 0, zeta6 - 1, zeta6, 0, -zeta6 + 1, -1]
sage: chi.conductor()
7
sage: chi.modulus()
21
sage: chi.order()
6
sage: chi(19)
-zeta6 + 1
sage: chi(40)
-zeta6 + 1
>>> from sage.all import *
>>> G = DirichletGroup(Integer(21))
>>> chi = G.gen(1); chi
Dirichlet character modulo 21 of conductor 7 mapping 8 |--> 1, 10 |--> zeta6
>>> chi.values()
[0, 1, zeta6 - 1, 0, -zeta6, -zeta6 + 1, 0, 0, 1, 0, zeta6, -zeta6, 0, -1,
0, 0, zeta6 - 1, zeta6, 0, -zeta6 + 1, -1]
>>> chi.conductor()
7
>>> chi.modulus()
21
>>> chi.order()
6
>>> chi(Integer(19))
-zeta6 + 1
>>> chi(Integer(40))
-zeta6 + 1
G = DirichletGroup(21) chi = G.1; chi chi.values() chi.conductor() chi.modulus() chi.order() chi(19) chi(40)
It is also possible to compute the action of the Galois group \(\text{Gal}(\QQ(\zeta_N)/\QQ)\) on these characters, as well as the direct product decomposition corresponding to the factorization of the modulus.
sage: chi.galois_orbit()
[Dirichlet character modulo 21 of conductor 7 mapping 8 |--> 1, 10 |--> -zeta6 + 1,
Dirichlet character modulo 21 of conductor 7 mapping 8 |--> 1, 10 |--> zeta6]
sage: go = G.galois_orbits()
sage: [len(orbit) for orbit in go]
[1, 2, 2, 1, 1, 2, 2, 1]
sage: G.decomposition()
[
Group of Dirichlet characters modulo 3 with values in Cyclotomic Field of order 6 and degree 2,
Group of Dirichlet characters modulo 7 with values in Cyclotomic Field of order 6 and degree 2
]
>>> from sage.all import *
>>> chi.galois_orbit()
[Dirichlet character modulo 21 of conductor 7 mapping 8 |--> 1, 10 |--> -zeta6 + 1,
Dirichlet character modulo 21 of conductor 7 mapping 8 |--> 1, 10 |--> zeta6]
>>> go = G.galois_orbits()
>>> [len(orbit) for orbit in go]
[1, 2, 2, 1, 1, 2, 2, 1]
>>> G.decomposition()
[
Group of Dirichlet characters modulo 3 with values in Cyclotomic Field of order 6 and degree 2,
Group of Dirichlet characters modulo 7 with values in Cyclotomic Field of order 6 and degree 2
]
chi.galois_orbit() go = G.galois_orbits() [len(orbit) for orbit in go] G.decomposition()
Next, we construct the group of Dirichlet characters mod 20, but with values in \(\QQ(i)\):
sage: K.<i> = NumberField(x^2+1)
sage: G = DirichletGroup(20,K)
sage: G
Group of Dirichlet characters modulo 20 with values in Number Field in i with defining polynomial x^2 + 1
>>> from sage.all import *
>>> K = NumberField(x**Integer(2)+Integer(1), names=('i',)); (i,) = K._first_ngens(1)
>>> G = DirichletGroup(Integer(20),K)
>>> G
Group of Dirichlet characters modulo 20 with values in Number Field in i with defining polynomial x^2 + 1
K.<i> = NumberField(x^2+1) G = DirichletGroup(20,K) G
We next compute several invariants of G:
sage: G.gens()
(Dirichlet character modulo 20 of conductor 4 mapping 11 |--> -1, 17 |--> 1,
Dirichlet character modulo 20 of conductor 5 mapping 11 |--> 1, 17 |--> i)
sage: G.unit_gens()
(11, 17)
sage: G.zeta()
i
sage: G.zeta_order()
4
>>> from sage.all import *
>>> G.gens()
(Dirichlet character modulo 20 of conductor 4 mapping 11 |--> -1, 17 |--> 1,
Dirichlet character modulo 20 of conductor 5 mapping 11 |--> 1, 17 |--> i)
>>> G.unit_gens()
(11, 17)
>>> G.zeta()
i
>>> G.zeta_order()
4
G.gens() G.unit_gens() G.zeta() G.zeta_order()
In this example we create a Dirichlet character with values in a
number field. We explicitly specify the choice of root of unity by
the third argument to DirichletGroup below.
sage: x = polygen(QQ, 'x')
sage: K = NumberField(x^4 + 1, 'a'); a = K.0
sage: b = K.gen(); a == b
True
sage: K
Number Field in a with defining polynomial x^4 + 1
sage: G = DirichletGroup(5, K, a); G
Group of Dirichlet characters modulo 5 with values in the group of order 8 generated by a in Number Field in a with defining polynomial x^4 + 1
sage: chi = G.0; chi
Dirichlet character modulo 5 of conductor 5 mapping 2 |--> a^2
sage: [(chi^i)(2) for i in range(4)]
[1, a^2, -1, -a^2]
>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(4) + Integer(1), 'a'); a = K.gen(0)
>>> b = K.gen(); a == b
True
>>> K
Number Field in a with defining polynomial x^4 + 1
>>> G = DirichletGroup(Integer(5), K, a); G
Group of Dirichlet characters modulo 5 with values in the group of order 8 generated by a in Number Field in a with defining polynomial x^4 + 1
>>> chi = G.gen(0); chi
Dirichlet character modulo 5 of conductor 5 mapping 2 |--> a^2
>>> [(chi**i)(Integer(2)) for i in range(Integer(4))]
[1, a^2, -1, -a^2]
x = polygen(QQ, 'x') K = NumberField(x^4 + 1, 'a'); a = K.0 b = K.gen(); a == b K G = DirichletGroup(5, K, a); G chi = G.0; chi [(chi^i)(2) for i in range(4)]
Here NumberField(x^4 + 1, 'a') tells Sage to use the symbol “a” in
printing what K is (a Number Field in a with defining polynomial
\(x^4 + 1\)). The name “a” is undeclared at this point. Once
a = K.0 (or equivalently a = K.gen()) is evaluated, the symbol
“a” represents a root of the generating polynomial
\(x^4+1\).
Modular Forms¶
Sage can do some computations related to modular forms, including dimensions, computing spaces of modular symbols, Hecke operators, and decompositions.
There are several functions available for computing dimensions of spaces of modular forms. For example,
sage: from sage.modular.dims import dimension_cusp_forms
sage: dimension_cusp_forms(Gamma0(11),2)
1
sage: dimension_cusp_forms(Gamma0(1),12)
1
sage: dimension_cusp_forms(Gamma1(389),2)
6112
>>> from sage.all import *
>>> from sage.modular.dims import dimension_cusp_forms
>>> dimension_cusp_forms(Gamma0(Integer(11)),Integer(2))
1
>>> dimension_cusp_forms(Gamma0(Integer(1)),Integer(12))
1
>>> dimension_cusp_forms(Gamma1(Integer(389)),Integer(2))
6112
from sage.modular.dims import dimension_cusp_forms dimension_cusp_forms(Gamma0(11),2) dimension_cusp_forms(Gamma0(1),12) dimension_cusp_forms(Gamma1(389),2)
Next we illustrate computation of Hecke operators on a space of modular symbols of level \(1\) and weight \(12\).
sage: M = ModularSymbols(1,12)
sage: M.basis()
([X^8*Y^2,(0,0)], [X^9*Y,(0,0)], [X^10,(0,0)])
sage: t2 = M.T(2)
sage: t2
Hecke operator T_2 on Modular Symbols space of dimension 3 for Gamma_0(1)
of weight 12 with sign 0 over Rational Field
sage: t2.matrix()
[ -24 0 0]
[ 0 -24 0]
[4860 0 2049]
sage: f = t2.charpoly('x'); f
x^3 - 2001*x^2 - 97776*x - 1180224
sage: factor(f)
(x - 2049) * (x + 24)^2
sage: M.T(11).charpoly('x').factor()
(x - 285311670612) * (x - 534612)^2
>>> from sage.all import *
>>> M = ModularSymbols(Integer(1),Integer(12))
>>> M.basis()
([X^8*Y^2,(0,0)], [X^9*Y,(0,0)], [X^10,(0,0)])
>>> t2 = M.T(Integer(2))
>>> t2
Hecke operator T_2 on Modular Symbols space of dimension 3 for Gamma_0(1)
of weight 12 with sign 0 over Rational Field
>>> t2.matrix()
[ -24 0 0]
[ 0 -24 0]
[4860 0 2049]
>>> f = t2.charpoly('x'); f
x^3 - 2001*x^2 - 97776*x - 1180224
>>> factor(f)
(x - 2049) * (x + 24)^2
>>> M.T(Integer(11)).charpoly('x').factor()
(x - 285311670612) * (x - 534612)^2
M = ModularSymbols(1,12)
M.basis()
t2 = M.T(2)
t2
t2.matrix()
f = t2.charpoly('x'); f
factor(f)
M.T(11).charpoly('x').factor()
We can also create spaces for \(\Gamma_0(N)\) and \(\Gamma_1(N)\).
sage: ModularSymbols(11,2)
Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign
0 over Rational Field
sage: ModularSymbols(Gamma1(11),2)
Modular Symbols space of dimension 11 for Gamma_1(11) of weight 2 with
sign 0 over Rational Field
>>> from sage.all import *
>>> ModularSymbols(Integer(11),Integer(2))
Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign
0 over Rational Field
>>> ModularSymbols(Gamma1(Integer(11)),Integer(2))
Modular Symbols space of dimension 11 for Gamma_1(11) of weight 2 with
sign 0 over Rational Field
ModularSymbols(11,2) ModularSymbols(Gamma1(11),2)
Let’s compute some characteristic polynomials and \(q\)-expansions.
sage: M = ModularSymbols(Gamma1(11),2)
sage: M.T(2).charpoly('x')
x^11 - 8*x^10 + 20*x^9 + 10*x^8 - 145*x^7 + 229*x^6 + 58*x^5 - 360*x^4
+ 70*x^3 - 515*x^2 + 1804*x - 1452
sage: M.T(2).charpoly('x').factor()
(x - 3) * (x + 2)^2 * (x^4 - 7*x^3 + 19*x^2 - 23*x + 11)
* (x^4 - 2*x^3 + 4*x^2 + 2*x + 11)
sage: S = M.cuspidal_submodule()
sage: S.T(2).matrix()
[-2 0]
[ 0 -2]
sage: S.q_expansion_basis(10)
[
q - 2*q^2 - q^3 + 2*q^4 + q^5 + 2*q^6 - 2*q^7 - 2*q^9 + O(q^10)
]
>>> from sage.all import *
>>> M = ModularSymbols(Gamma1(Integer(11)),Integer(2))
>>> M.T(Integer(2)).charpoly('x')
x^11 - 8*x^10 + 20*x^9 + 10*x^8 - 145*x^7 + 229*x^6 + 58*x^5 - 360*x^4
+ 70*x^3 - 515*x^2 + 1804*x - 1452
>>> M.T(Integer(2)).charpoly('x').factor()
(x - 3) * (x + 2)^2 * (x^4 - 7*x^3 + 19*x^2 - 23*x + 11)
* (x^4 - 2*x^3 + 4*x^2 + 2*x + 11)
>>> S = M.cuspidal_submodule()
>>> S.T(Integer(2)).matrix()
[-2 0]
[ 0 -2]
>>> S.q_expansion_basis(Integer(10))
[
q - 2*q^2 - q^3 + 2*q^4 + q^5 + 2*q^6 - 2*q^7 - 2*q^9 + O(q^10)
]
M = ModularSymbols(Gamma1(11),2)
M.T(2).charpoly('x')
M.T(2).charpoly('x').factor()
S = M.cuspidal_submodule()
S.T(2).matrix()
S.q_expansion_basis(10)
We can even compute spaces of modular symbols with character.
sage: G = DirichletGroup(13)
sage: e = G.0^2
sage: M = ModularSymbols(e,2); M
Modular Symbols space of dimension 4 and level 13, weight 2, character
[zeta6], sign 0, over Cyclotomic Field of order 6 and degree 2
sage: M.T(2).charpoly('x').factor()
(x - zeta6 - 2) * (x - 2*zeta6 - 1) * (x + zeta6 + 1)^2
sage: S = M.cuspidal_submodule(); S
Modular Symbols subspace of dimension 2 of Modular Symbols space of
dimension 4 and level 13, weight 2, character [zeta6], sign 0, over
Cyclotomic Field of order 6 and degree 2
sage: S.T(2).charpoly('x').factor()
(x + zeta6 + 1)^2
sage: S.q_expansion_basis(10)
[
q + (-zeta6 - 1)*q^2 + (2*zeta6 - 2)*q^3 + zeta6*q^4 + (-2*zeta6 + 1)*q^5
+ (-2*zeta6 + 4)*q^6 + (2*zeta6 - 1)*q^8 - zeta6*q^9 + O(q^10)
]
>>> from sage.all import *
>>> G = DirichletGroup(Integer(13))
>>> e = G.gen(0)**Integer(2)
>>> M = ModularSymbols(e,Integer(2)); M
Modular Symbols space of dimension 4 and level 13, weight 2, character
[zeta6], sign 0, over Cyclotomic Field of order 6 and degree 2
>>> M.T(Integer(2)).charpoly('x').factor()
(x - zeta6 - 2) * (x - 2*zeta6 - 1) * (x + zeta6 + 1)^2
>>> S = M.cuspidal_submodule(); S
Modular Symbols subspace of dimension 2 of Modular Symbols space of
dimension 4 and level 13, weight 2, character [zeta6], sign 0, over
Cyclotomic Field of order 6 and degree 2
>>> S.T(Integer(2)).charpoly('x').factor()
(x + zeta6 + 1)^2
>>> S.q_expansion_basis(Integer(10))
[
q + (-zeta6 - 1)*q^2 + (2*zeta6 - 2)*q^3 + zeta6*q^4 + (-2*zeta6 + 1)*q^5
+ (-2*zeta6 + 4)*q^6 + (2*zeta6 - 1)*q^8 - zeta6*q^9 + O(q^10)
]
G = DirichletGroup(13)
e = G.0^2
M = ModularSymbols(e,2); M
M.T(2).charpoly('x').factor()
S = M.cuspidal_submodule(); S
S.T(2).charpoly('x').factor()
S.q_expansion_basis(10)
Here is another example of how Sage can compute the action of Hecke operators on a space of modular forms.
sage: T = ModularForms(Gamma0(11),2)
sage: T
Modular Forms space of dimension 2 for Congruence Subgroup Gamma0(11) of
weight 2 over Rational Field
sage: T.degree()
2
sage: T.level()
11
sage: T.group()
Congruence Subgroup Gamma0(11)
sage: T.dimension()
2
sage: T.cuspidal_subspace()
Cuspidal subspace of dimension 1 of Modular Forms space of dimension 2 for
Congruence Subgroup Gamma0(11) of weight 2 over Rational Field
sage: T.eisenstein_subspace()
Eisenstein subspace of dimension 1 of Modular Forms space of dimension 2
for Congruence Subgroup Gamma0(11) of weight 2 over Rational Field
sage: M = ModularSymbols(11); M
Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign
0 over Rational Field
sage: M.weight()
2
sage: M.basis()
((1,0), (1,8), (1,9))
sage: M.sign()
0
>>> from sage.all import *
>>> T = ModularForms(Gamma0(Integer(11)),Integer(2))
>>> T
Modular Forms space of dimension 2 for Congruence Subgroup Gamma0(11) of
weight 2 over Rational Field
>>> T.degree()
2
>>> T.level()
11
>>> T.group()
Congruence Subgroup Gamma0(11)
>>> T.dimension()
2
>>> T.cuspidal_subspace()
Cuspidal subspace of dimension 1 of Modular Forms space of dimension 2 for
Congruence Subgroup Gamma0(11) of weight 2 over Rational Field
>>> T.eisenstein_subspace()
Eisenstein subspace of dimension 1 of Modular Forms space of dimension 2
for Congruence Subgroup Gamma0(11) of weight 2 over Rational Field
>>> M = ModularSymbols(Integer(11)); M
Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign
0 over Rational Field
>>> M.weight()
2
>>> M.basis()
((1,0), (1,8), (1,9))
>>> M.sign()
0
T = ModularForms(Gamma0(11),2) T T.degree() T.level() T.group() T.dimension() T.cuspidal_subspace() T.eisenstein_subspace() M = ModularSymbols(11); M M.weight() M.basis() M.sign()
Let \(T_p\) denote the usual Hecke operators (\(p\) prime). How do the Hecke operators \(T_2\), \(T_3\), \(T_5\) act on the space of modular symbols?
sage: M.T(2).matrix()
[ 3 0 -1]
[ 0 -2 0]
[ 0 0 -2]
sage: M.T(3).matrix()
[ 4 0 -1]
[ 0 -1 0]
[ 0 0 -1]
sage: M.T(5).matrix()
[ 6 0 -1]
[ 0 1 0]
[ 0 0 1]
>>> from sage.all import *
>>> M.T(Integer(2)).matrix()
[ 3 0 -1]
[ 0 -2 0]
[ 0 0 -2]
>>> M.T(Integer(3)).matrix()
[ 4 0 -1]
[ 0 -1 0]
[ 0 0 -1]
>>> M.T(Integer(5)).matrix()
[ 6 0 -1]
[ 0 1 0]
[ 0 0 1]
M.T(2).matrix() M.T(3).matrix() M.T(5).matrix()