Hyperelliptic curves over a finite field¶

EXAMPLES:

Sage

sage: K.<a> = GF(9, 'a')
sage: x = polygen(K)
sage: C = HyperellipticCurve(x^7 - x^5 - 2, x^2 + a)
sage: C._points_fast_sqrt()
[(0 : 1 : 0), (a + 1 : a : 1), (a + 1 : a + 1 : 1), (2 : a + 1 : 1),
 (2*a : 2*a + 2 : 1), (2*a : 2*a : 1), (1 : a + 1 : 1)]

Python

>>> from sage.all import *
>>> K = GF(Integer(9), 'a', names=('a',)); (a,) = K._first_ngens(1)
>>> x = polygen(K)
>>> C = HyperellipticCurve(x**Integer(7) - x**Integer(5) - Integer(2), x**Integer(2) + a)
>>> C._points_fast_sqrt()
[(0 : 1 : 0), (a + 1 : a : 1), (a + 1 : a + 1 : 1), (2 : a + 1 : 1),
 (2*a : 2*a + 2 : 1), (2*a : 2*a : 1), (1 : a + 1 : 1)]

Sage Live

K.<a> = GF(9, 'a')
x = polygen(K)
C = HyperellipticCurve(x^7 - x^5 - 2, x^2 + a)
C._points_fast_sqrt()

AUTHORS:

David Kohel (2006)
Robert Bradshaw (2007)
Alyson Deines, Marina Gresham, Gagan Sekhon, (2010)
Daniel Krenn (2011)
Jean-Pierre Flori, Jan Tuitman (2013)
Kiran Kedlaya (2016)
Dean Bisogno (2017): Fixed Hasse-Witt computation

class sage.schemes.hyperelliptic_curves.hyperelliptic_finite_field.HyperellipticCurve_finite_field(PP, f, h=None, names=None, genus=None)[source]¶

Bases: HyperellipticCurve_generic, ProjectivePlaneCurve_finite_field

Cartier_matrix()[source]¶

INPUT:

self – Hyperelliptic Curve of the form \(y^2 = f(x)\) over a finite field, \(\GF{q}\)

OUTPUT:

The matrix \(M = (c_{pi-j})\), where \(c_i\) are the coefficients of \(f(x)^{(p-1)/2} = \sum c_i x^i\).

REFERENCES:

Yui. On the Jacobian varieties of hyperelliptic curves over fields of characteristic \(p > 2\).

EXAMPLES:

Sage

sage: K.<x> = GF(9,'x')[]
sage: C = HyperellipticCurve(x^7 - 1, 0)
sage: C.Cartier_matrix()
[0 0 2]
[0 0 0]
[0 1 0]

sage: K.<x> = GF(49, 'x')[]
sage: C = HyperellipticCurve(x^5 + 1, 0)
sage: C.Cartier_matrix()
[0 3]
[0 0]

sage: P.<x> = GF(9, 'a')[]
sage: E = HyperellipticCurve(x^29 + 1, 0)
sage: E.Cartier_matrix()
[0 0 1 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 1 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 1 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 1 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[1 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 1 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 1 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 1 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 1 0]

Python

>>> from sage.all import *
>>> K = GF(Integer(9),'x')['x']; (x,) = K._first_ngens(1)
>>> C = HyperellipticCurve(x**Integer(7) - Integer(1), Integer(0))
>>> C.Cartier_matrix()
[0 0 2]
[0 0 0]
[0 1 0]

>>> K = GF(Integer(49), 'x')['x']; (x,) = K._first_ngens(1)
>>> C = HyperellipticCurve(x**Integer(5) + Integer(1), Integer(0))
>>> C.Cartier_matrix()
[0 3]
[0 0]

>>> P = GF(Integer(9), 'a')['x']; (x,) = P._first_ngens(1)
>>> E = HyperellipticCurve(x**Integer(29) + Integer(1), Integer(0))
>>> E.Cartier_matrix()
[0 0 1 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 1 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 1 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 1 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[1 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 1 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 1 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 1 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 1 0]

Sage Live

K.<x> = GF(9,'x')[]
C = HyperellipticCurve(x^7 - 1, 0)
C.Cartier_matrix()
K.<x> = GF(49, 'x')[]
C = HyperellipticCurve(x^5 + 1, 0)
C.Cartier_matrix()
P.<x> = GF(9, 'a')[]
E = HyperellipticCurve(x^29 + 1, 0)
E.Cartier_matrix()

Hasse_Witt()[source]¶

INPUT:

self – Hyperelliptic Curve of the form \(y^2 = f(x)\) over a finite field, \(\GF{q}\)

OUTPUT:

The matrix \(N = M M^p \dots M^{p^{g-1}}\) where \(M = c_{pi-j}\), and \(f(x)^{(p-1)/2} = \sum c_i x^i\).

Reference-N. Yui. On the Jacobian varieties of hyperelliptic curves over fields of characteristic \(p > 2\).

EXAMPLES:

Sage

sage: K.<x> = GF(9, 'x')[]
sage: C = HyperellipticCurve(x^7 - 1, 0)
sage: C.Hasse_Witt()
[0 0 0]
[0 0 0]
[0 0 0]

sage: K.<x> = GF(49, 'x')[]
sage: C = HyperellipticCurve(x^5 + 1, 0)
sage: C.Hasse_Witt()
[0 0]
[0 0]

sage: P.<x> = GF(9, 'a')[]
sage: E = HyperellipticCurve(x^29 + 1, 0)
sage: E.Hasse_Witt()
[0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0]

Python

>>> from sage.all import *
>>> K = GF(Integer(9), 'x')['x']; (x,) = K._first_ngens(1)
>>> C = HyperellipticCurve(x**Integer(7) - Integer(1), Integer(0))
>>> C.Hasse_Witt()
[0 0 0]
[0 0 0]
[0 0 0]

>>> K = GF(Integer(49), 'x')['x']; (x,) = K._first_ngens(1)
>>> C = HyperellipticCurve(x**Integer(5) + Integer(1), Integer(0))
>>> C.Hasse_Witt()
[0 0]
[0 0]

>>> P = GF(Integer(9), 'a')['x']; (x,) = P._first_ngens(1)
>>> E = HyperellipticCurve(x**Integer(29) + Integer(1), Integer(0))
>>> E.Hasse_Witt()
[0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0]

Sage Live

K.<x> = GF(9, 'x')[]
C = HyperellipticCurve(x^7 - 1, 0)
C.Hasse_Witt()
K.<x> = GF(49, 'x')[]
C = HyperellipticCurve(x^5 + 1, 0)
C.Hasse_Witt()
P.<x> = GF(9, 'a')[]
E = HyperellipticCurve(x^29 + 1, 0)
E.Hasse_Witt()

a_number()[source]¶

INPUT:

self – Hyperelliptic Curve of the form \(y^2 = f(x)\) over a finite field, \(\GF{q}\)

OUTPUT: a-number

EXAMPLES:

Sage

sage: K.<x> = GF(49, 'x')[]
sage: C = HyperellipticCurve(x^5 + 1, 0)
sage: C.a_number()
1

sage: K.<x> = GF(9, 'x')[]
sage: C = HyperellipticCurve(x^7 - 1, 0)
sage: C.a_number()
1

sage: P.<x> = GF(9, 'a')[]
sage: E = HyperellipticCurve(x^29 + 1, 0)
sage: E.a_number()
5

Python

>>> from sage.all import *
>>> K = GF(Integer(49), 'x')['x']; (x,) = K._first_ngens(1)
>>> C = HyperellipticCurve(x**Integer(5) + Integer(1), Integer(0))
>>> C.a_number()
1

>>> K = GF(Integer(9), 'x')['x']; (x,) = K._first_ngens(1)
>>> C = HyperellipticCurve(x**Integer(7) - Integer(1), Integer(0))
>>> C.a_number()
1

>>> P = GF(Integer(9), 'a')['x']; (x,) = P._first_ngens(1)
>>> E = HyperellipticCurve(x**Integer(29) + Integer(1), Integer(0))
>>> E.a_number()
5

Sage Live

K.<x> = GF(49, 'x')[]
C = HyperellipticCurve(x^5 + 1, 0)
C.a_number()
K.<x> = GF(9, 'x')[]
C = HyperellipticCurve(x^7 - 1, 0)
C.a_number()
P.<x> = GF(9, 'a')[]
E = HyperellipticCurve(x^29 + 1, 0)
E.a_number()

cardinality(extension_degree=1)[source]¶

Count points on a single extension of the base field.

EXAMPLES:

Sage

sage: K = GF(101)
sage: R.<t> = PolynomialRing(K)
sage: H = HyperellipticCurve(t^9 + 3*t^5 + 5)
sage: H.cardinality()
106
sage: H.cardinality(15)
1160968955369992567076405831000
sage: H.cardinality(100)
270481382942152609326719471080753083367793838278100277689020104911710151430673927943945601434674459120495370826289654897190781715493352266982697064575800553229661690000887425442240414673923744999504000

sage: K = GF(37)
sage: R.<t> = PolynomialRing(K)
sage: H = HyperellipticCurve(t^9 + 3*t^5 + 5)
sage: H.cardinality()
40
sage: H.cardinality(2)
1408
sage: H.cardinality(3)
50116

Python

>>> from sage.all import *
>>> K = GF(Integer(101))
>>> R = PolynomialRing(K, names=('t',)); (t,) = R._first_ngens(1)
>>> H = HyperellipticCurve(t**Integer(9) + Integer(3)*t**Integer(5) + Integer(5))
>>> H.cardinality()
106
>>> H.cardinality(Integer(15))
1160968955369992567076405831000
>>> H.cardinality(Integer(100))
270481382942152609326719471080753083367793838278100277689020104911710151430673927943945601434674459120495370826289654897190781715493352266982697064575800553229661690000887425442240414673923744999504000

>>> K = GF(Integer(37))
>>> R = PolynomialRing(K, names=('t',)); (t,) = R._first_ngens(1)
>>> H = HyperellipticCurve(t**Integer(9) + Integer(3)*t**Integer(5) + Integer(5))
>>> H.cardinality()
40
>>> H.cardinality(Integer(2))
1408
>>> H.cardinality(Integer(3))
50116

Sage Live

K = GF(101)
R.<t> = PolynomialRing(K)
H = HyperellipticCurve(t^9 + 3*t^5 + 5)
H.cardinality()
H.cardinality(15)
H.cardinality(100)
K = GF(37)
R.<t> = PolynomialRing(K)
H = HyperellipticCurve(t^9 + 3*t^5 + 5)
H.cardinality()
H.cardinality(2)
H.cardinality(3)

The following example shows that Issue #20391 has been resolved:

Sage

sage: F=GF(23)
sage: x=polygen(F)
sage: C=HyperellipticCurve(x^8+1)
sage: C.cardinality()
24

Python

>>> from sage.all import *
>>> F=GF(Integer(23))
>>> x=polygen(F)
>>> C=HyperellipticCurve(x**Integer(8)+Integer(1))
>>> C.cardinality()
24

Sage Live

F=GF(23)
x=polygen(F)
C=HyperellipticCurve(x^8+1)
C.cardinality()

cardinality_exhaustive(extension_degree=1, algorithm=None)[source]¶

Count points on a single extension of the base field by enumerating over x and solving the resulting quadratic equation for y.

EXAMPLES:

Sage

sage: K.<a> = GF(9, 'a')
sage: x = polygen(K)
sage: C = HyperellipticCurve(x^7 - 1, x^2 + a)
sage: C.cardinality_exhaustive()
7

sage: K = GF(next_prime(1<<10))
sage: R.<t> = PolynomialRing(K)
sage: H = HyperellipticCurve(t^7 + 3*t^5 + 5)
sage: H.cardinality_exhaustive()
1025

sage: P.<x> = PolynomialRing(GF(9,'a'))
sage: H = HyperellipticCurve(x^5+x^2+1)
sage: H.count_points(5)
[18, 78, 738, 6366, 60018]

sage: F.<a> = GF(4); P.<x> = F[]
sage: H = HyperellipticCurve(x^5+a*x^2+1, x+a+1)
sage: H.count_points(6)
[2, 24, 74, 256, 1082, 4272]

Python

>>> from sage.all import *
>>> K = GF(Integer(9), 'a', names=('a',)); (a,) = K._first_ngens(1)
>>> x = polygen(K)
>>> C = HyperellipticCurve(x**Integer(7) - Integer(1), x**Integer(2) + a)
>>> C.cardinality_exhaustive()
7

>>> K = GF(next_prime(Integer(1)<<Integer(10)))
>>> R = PolynomialRing(K, names=('t',)); (t,) = R._first_ngens(1)
>>> H = HyperellipticCurve(t**Integer(7) + Integer(3)*t**Integer(5) + Integer(5))
>>> H.cardinality_exhaustive()
1025

>>> P = PolynomialRing(GF(Integer(9),'a'), names=('x',)); (x,) = P._first_ngens(1)
>>> H = HyperellipticCurve(x**Integer(5)+x**Integer(2)+Integer(1))
>>> H.count_points(Integer(5))
[18, 78, 738, 6366, 60018]

>>> F = GF(Integer(4), names=('a',)); (a,) = F._first_ngens(1); P = F['x']; (x,) = P._first_ngens(1)
>>> H = HyperellipticCurve(x**Integer(5)+a*x**Integer(2)+Integer(1), x+a+Integer(1))
>>> H.count_points(Integer(6))
[2, 24, 74, 256, 1082, 4272]

Sage Live

K.<a> = GF(9, 'a')
x = polygen(K)
C = HyperellipticCurve(x^7 - 1, x^2 + a)
C.cardinality_exhaustive()
K = GF(next_prime(1<<10))
R.<t> = PolynomialRing(K)
H = HyperellipticCurve(t^7 + 3*t^5 + 5)
H.cardinality_exhaustive()
P.<x> = PolynomialRing(GF(9,'a'))
H = HyperellipticCurve(x^5+x^2+1)
H.count_points(5)
F.<a> = GF(4); P.<x> = F[]
H = HyperellipticCurve(x^5+a*x^2+1, x+a+1)
H.count_points(6)

cardinality_hypellfrob(extension_degree=1, algorithm=None)[source]¶

Count points on a single extension of the base field using the hypellfrob program.

EXAMPLES:

Sage

sage: K = GF(next_prime(1<<10))
sage: R.<t> = PolynomialRing(K)
sage: H = HyperellipticCurve(t^7 + 3*t^5 + 5)
sage: H.cardinality_hypellfrob()
1025

sage: K = GF(49999)
sage: R.<t> = PolynomialRing(K)
sage: H = HyperellipticCurve(t^7 + 3*t^5 + 5)
sage: H.cardinality_hypellfrob()
50162
sage: H.cardinality_hypellfrob(3)
124992471088310

Python

>>> from sage.all import *
>>> K = GF(next_prime(Integer(1)<<Integer(10)))
>>> R = PolynomialRing(K, names=('t',)); (t,) = R._first_ngens(1)
>>> H = HyperellipticCurve(t**Integer(7) + Integer(3)*t**Integer(5) + Integer(5))
>>> H.cardinality_hypellfrob()
1025

>>> K = GF(Integer(49999))
>>> R = PolynomialRing(K, names=('t',)); (t,) = R._first_ngens(1)
>>> H = HyperellipticCurve(t**Integer(7) + Integer(3)*t**Integer(5) + Integer(5))
>>> H.cardinality_hypellfrob()
50162
>>> H.cardinality_hypellfrob(Integer(3))
124992471088310

Sage Live

K = GF(next_prime(1<<10))
R.<t> = PolynomialRing(K)
H = HyperellipticCurve(t^7 + 3*t^5 + 5)
H.cardinality_hypellfrob()
K = GF(49999)
R.<t> = PolynomialRing(K)
H = HyperellipticCurve(t^7 + 3*t^5 + 5)
H.cardinality_hypellfrob()
H.cardinality_hypellfrob(3)

count_points(n=1)[source]¶

Count points over finite fields.

INPUT:

n – integer

OUTPUT:

An integer. The number of points over \(\GF{q}, \ldots, \GF{q^n}\) on a hyperelliptic curve over a finite field \(\GF{q}\).

Warning

This is currently using exhaustive search for hyperelliptic curves over non-prime fields, which can be awfully slow.

EXAMPLES:

Sage

sage: P.<x> = PolynomialRing(GF(3))
sage: C = HyperellipticCurve(x^3+x^2+1)
sage: C.count_points(4)
[6, 12, 18, 96]
sage: C.base_extend(GF(9,'a')).count_points(2)
[12, 96]

sage: K = GF(2**31-1)
sage: R.<t> = PolynomialRing(K)
sage: H = HyperellipticCurve(t^5 + 3*t + 5)
sage: H.count_points() # long time, 2.4 sec on a Corei7
[2147464821]
sage: H.count_points(n=2) # long time, 30s on a Corei7
[2147464821, 4611686018988310237]

sage: K = GF(2**7-1)
sage: R.<t> = PolynomialRing(K)
sage: H = HyperellipticCurve(t^13 + 3*t^5 + 5)
sage: H.count_points(n=6)
[112, 16360, 2045356, 260199160, 33038302802, 4195868633548]

sage: P.<x> = PolynomialRing(GF(3))
sage: H = HyperellipticCurve(x^3+x^2+1)
sage: C1 = H.count_points(4); C1
[6, 12, 18, 96]
sage: C2 = sage.schemes.generic.scheme.Scheme.count_points(H,4); C2 # long time, 2s on a Corei7
[6, 12, 18, 96]
sage: C1 == C2 # long time, because we need C2 to be defined
True

sage: P.<x> = PolynomialRing(GF(9,'a'))
sage: H = HyperellipticCurve(x^5+x^2+1)
sage: H.count_points(5)
[18, 78, 738, 6366, 60018]

sage: F.<a> = GF(4); P.<x> = F[]
sage: H = HyperellipticCurve(x^5+a*x^2+1, x+a+1)
sage: H.count_points(6)
[2, 24, 74, 256, 1082, 4272]

Python

>>> from sage.all import *
>>> P = PolynomialRing(GF(Integer(3)), names=('x',)); (x,) = P._first_ngens(1)
>>> C = HyperellipticCurve(x**Integer(3)+x**Integer(2)+Integer(1))
>>> C.count_points(Integer(4))
[6, 12, 18, 96]
>>> C.base_extend(GF(Integer(9),'a')).count_points(Integer(2))
[12, 96]

>>> K = GF(Integer(2)**Integer(31)-Integer(1))
>>> R = PolynomialRing(K, names=('t',)); (t,) = R._first_ngens(1)
>>> H = HyperellipticCurve(t**Integer(5) + Integer(3)*t + Integer(5))
>>> H.count_points() # long time, 2.4 sec on a Corei7
[2147464821]
>>> H.count_points(n=Integer(2)) # long time, 30s on a Corei7
[2147464821, 4611686018988310237]

>>> K = GF(Integer(2)**Integer(7)-Integer(1))
>>> R = PolynomialRing(K, names=('t',)); (t,) = R._first_ngens(1)
>>> H = HyperellipticCurve(t**Integer(13) + Integer(3)*t**Integer(5) + Integer(5))
>>> H.count_points(n=Integer(6))
[112, 16360, 2045356, 260199160, 33038302802, 4195868633548]

>>> P = PolynomialRing(GF(Integer(3)), names=('x',)); (x,) = P._first_ngens(1)
>>> H = HyperellipticCurve(x**Integer(3)+x**Integer(2)+Integer(1))
>>> C1 = H.count_points(Integer(4)); C1
[6, 12, 18, 96]
>>> C2 = sage.schemes.generic.scheme.Scheme.count_points(H,Integer(4)); C2 # long time, 2s on a Corei7
[6, 12, 18, 96]
>>> C1 == C2 # long time, because we need C2 to be defined
True

>>> P = PolynomialRing(GF(Integer(9),'a'), names=('x',)); (x,) = P._first_ngens(1)
>>> H = HyperellipticCurve(x**Integer(5)+x**Integer(2)+Integer(1))
>>> H.count_points(Integer(5))
[18, 78, 738, 6366, 60018]

>>> F = GF(Integer(4), names=('a',)); (a,) = F._first_ngens(1); P = F['x']; (x,) = P._first_ngens(1)
>>> H = HyperellipticCurve(x**Integer(5)+a*x**Integer(2)+Integer(1), x+a+Integer(1))
>>> H.count_points(Integer(6))
[2, 24, 74, 256, 1082, 4272]

Sage Live

P.<x> = PolynomialRing(GF(3))
C = HyperellipticCurve(x^3+x^2+1)
C.count_points(4)
C.base_extend(GF(9,'a')).count_points(2)
K = GF(2**31-1)
R.<t> = PolynomialRing(K)
H = HyperellipticCurve(t^5 + 3*t + 5)
H.count_points() # long time, 2.4 sec on a Corei7
H.count_points(n=2) # long time, 30s on a Corei7
K = GF(2**7-1)
R.<t> = PolynomialRing(K)
H = HyperellipticCurve(t^13 + 3*t^5 + 5)
H.count_points(n=6)
P.<x> = PolynomialRing(GF(3))
H = HyperellipticCurve(x^3+x^2+1)
C1 = H.count_points(4); C1
C2 = sage.schemes.generic.scheme.Scheme.count_points(H,4); C2 # long time, 2s on a Corei7
C1 == C2 # long time, because we need C2 to be defined
P.<x> = PolynomialRing(GF(9,'a'))
H = HyperellipticCurve(x^5+x^2+1)
H.count_points(5)
F.<a> = GF(4); P.<x> = F[]
H = HyperellipticCurve(x^5+a*x^2+1, x+a+1)
H.count_points(6)

This example shows that Issue #20391 is resolved:

Sage

sage: x = polygen(GF(4099))
sage: H = HyperellipticCurve(x^6 + x + 1)
sage: H.count_points(1)
[4106]

Python

>>> from sage.all import *
>>> x = polygen(GF(Integer(4099)))
>>> H = HyperellipticCurve(x**Integer(6) + x + Integer(1))
>>> H.count_points(Integer(1))
[4106]

Sage Live

x = polygen(GF(4099))
H = HyperellipticCurve(x^6 + x + 1)
H.count_points(1)

count_points_exhaustive(n=1, naive=False)[source]¶

Count the number of points on the curve over the first \(n\) extensions of the base field by exhaustive search if \(n\) if smaller than \(g\), the genus of the curve, and by computing the frobenius polynomial after performing exhaustive search on the first \(g\) extensions if \(n > g\) (unless naive == True).

EXAMPLES:

Sage

sage: K = GF(5)
sage: R.<t> = PolynomialRing(K)
sage: H = HyperellipticCurve(t^9 + t^3 + 1)
sage: H.count_points_exhaustive(n=5)
[9, 27, 108, 675, 3069]

Python

>>> from sage.all import *
>>> K = GF(Integer(5))
>>> R = PolynomialRing(K, names=('t',)); (t,) = R._first_ngens(1)
>>> H = HyperellipticCurve(t**Integer(9) + t**Integer(3) + Integer(1))
>>> H.count_points_exhaustive(n=Integer(5))
[9, 27, 108, 675, 3069]

Sage Live

K = GF(5)
R.<t> = PolynomialRing(K)
H = HyperellipticCurve(t^9 + t^3 + 1)
H.count_points_exhaustive(n=5)

When \(n > g\), the frobenius polynomial is computed from the numbers of points of the curve over the first \(g\) extension, so that computing the number of points on extensions of degree \(n > g\) is not much more expensive than for \(n == g\):

Sage

sage: H.count_points_exhaustive(n=15)
[9,
27,
108,
675,
3069,
16302,
78633,
389475,
1954044,
9768627,
48814533,
244072650,
1220693769,
6103414827,
30517927308]

Python

>>> from sage.all import *
>>> H.count_points_exhaustive(n=Integer(15))
[9,
27,
108,
675,
3069,
16302,
78633,
389475,
1954044,
9768627,
48814533,
244072650,
1220693769,
6103414827,
30517927308]

Sage Live

H.count_points_exhaustive(n=15)

This behavior can be disabled by passing naive=True:

Sage

sage: H.count_points_exhaustive(n=6, naive=True) # long time, 7s on a Corei7
[9, 27, 108, 675, 3069, 16302]

Python

>>> from sage.all import *
>>> H.count_points_exhaustive(n=Integer(6), naive=True) # long time, 7s on a Corei7
[9, 27, 108, 675, 3069, 16302]

Sage Live

H.count_points_exhaustive(n=6, naive=True) # long time, 7s on a Corei7

count_points_frobenius_polynomial(n=1, f=None)[source]¶

Count the number of points on the curve over the first \(n\) extensions of the base field by computing the frobenius polynomial.

EXAMPLES:

Sage

sage: K = GF(49999)
sage: R.<t> = PolynomialRing(K)
sage: H = HyperellipticCurve(t^19 + t + 1)

Python

>>> from sage.all import *
>>> K = GF(Integer(49999))
>>> R = PolynomialRing(K, names=('t',)); (t,) = R._first_ngens(1)
>>> H = HyperellipticCurve(t**Integer(19) + t + Integer(1))

Sage Live

K = GF(49999)
R.<t> = PolynomialRing(K)
H = HyperellipticCurve(t^19 + t + 1)

The following computation takes a long time as the complete characteristic polynomial of the frobenius is computed:

Sage

sage: H.count_points_frobenius_polynomial(3) # long time, 20s on a Corei7 (when computed before the following test of course)
[49491, 2500024375, 124992509154249]

Python

>>> from sage.all import *
>>> H.count_points_frobenius_polynomial(Integer(3)) # long time, 20s on a Corei7 (when computed before the following test of course)
[49491, 2500024375, 124992509154249]

Sage Live

H.count_points_frobenius_polynomial(3) # long time, 20s on a Corei7 (when computed before the following test of course)

As the polynomial is cached, further computations of number of points are really fast:

Sage

sage: H.count_points_frobenius_polynomial(19) # long time, because of the previous test
[49491,
2500024375,
124992509154249,
6249500007135192947,
312468751250758776051811,
15623125093747382662737313867,
781140631562281338861289572576257,
39056250437482500417107992413002794587,
1952773465623687539373429411200893147181079,
97636720507718753281169963459063147221761552935,
4881738388665429945305281187129778704058864736771824,
244082037694882831835318764490138139735446240036293092851,
12203857802706446708934102903106811520015567632046432103159713,
610180686277519628999996211052002771035439565767719719151141201339,
30508424133189703930370810556389262704405225546438978173388673620145499,
1525390698235352006814610157008906752699329454643826047826098161898351623931,
76268009521069364988723693240288328729528917832735078791261015331201838856825193,
3813324208043947180071195938321176148147244128062172555558715783649006587868272993991,
190662397077989315056379725720120486231213267083935859751911720230901597698389839098903847]

Python

>>> from sage.all import *
>>> H.count_points_frobenius_polynomial(Integer(19)) # long time, because of the previous test
[49491,
2500024375,
124992509154249,
6249500007135192947,
312468751250758776051811,
15623125093747382662737313867,
781140631562281338861289572576257,
39056250437482500417107992413002794587,
1952773465623687539373429411200893147181079,
97636720507718753281169963459063147221761552935,
4881738388665429945305281187129778704058864736771824,
244082037694882831835318764490138139735446240036293092851,
12203857802706446708934102903106811520015567632046432103159713,
610180686277519628999996211052002771035439565767719719151141201339,
30508424133189703930370810556389262704405225546438978173388673620145499,
1525390698235352006814610157008906752699329454643826047826098161898351623931,
76268009521069364988723693240288328729528917832735078791261015331201838856825193,
3813324208043947180071195938321176148147244128062172555558715783649006587868272993991,
190662397077989315056379725720120486231213267083935859751911720230901597698389839098903847]

Sage Live

H.count_points_frobenius_polynomial(19) # long time, because of the previous test

count_points_hypellfrob(n=1, N=None, algorithm=None)[source]¶

Count the number of points on the curve over the first \(n\) extensions of the base field using the hypellfrob program.

This only supports prime fields of large enough characteristic.

EXAMPLES:

Sage

sage: K = GF(49999)
sage: R.<t> = PolynomialRing(K)
sage: H = HyperellipticCurve(t^21 + 3*t^5 + 5)
sage: H.count_points_hypellfrob()
[49804]
sage: H.count_points_hypellfrob(2)
[49804, 2499799038]

sage: K = GF(2**7-1)
sage: R.<t> = PolynomialRing(K)
sage: H = HyperellipticCurve(t^11 + 3*t^5 + 5)
sage: H.count_points_hypellfrob()
[127]
sage: H.count_points_hypellfrob(n=5)
[127, 16335, 2045701, 260134299, 33038098487]

sage: K = GF(2**7-1)
sage: R.<t> = PolynomialRing(K)
sage: H = HyperellipticCurve(t^13 + 3*t^5 + 5)
sage: H.count_points(n=6)
[112, 16360, 2045356, 260199160, 33038302802, 4195868633548]

Python

>>> from sage.all import *
>>> K = GF(Integer(49999))
>>> R = PolynomialRing(K, names=('t',)); (t,) = R._first_ngens(1)
>>> H = HyperellipticCurve(t**Integer(21) + Integer(3)*t**Integer(5) + Integer(5))
>>> H.count_points_hypellfrob()
[49804]
>>> H.count_points_hypellfrob(Integer(2))
[49804, 2499799038]

>>> K = GF(Integer(2)**Integer(7)-Integer(1))
>>> R = PolynomialRing(K, names=('t',)); (t,) = R._first_ngens(1)
>>> H = HyperellipticCurve(t**Integer(11) + Integer(3)*t**Integer(5) + Integer(5))
>>> H.count_points_hypellfrob()
[127]
>>> H.count_points_hypellfrob(n=Integer(5))
[127, 16335, 2045701, 260134299, 33038098487]

>>> K = GF(Integer(2)**Integer(7)-Integer(1))
>>> R = PolynomialRing(K, names=('t',)); (t,) = R._first_ngens(1)
>>> H = HyperellipticCurve(t**Integer(13) + Integer(3)*t**Integer(5) + Integer(5))
>>> H.count_points(n=Integer(6))
[112, 16360, 2045356, 260199160, 33038302802, 4195868633548]

Sage Live

K = GF(49999)
R.<t> = PolynomialRing(K)
H = HyperellipticCurve(t^21 + 3*t^5 + 5)
H.count_points_hypellfrob()
H.count_points_hypellfrob(2)
K = GF(2**7-1)
R.<t> = PolynomialRing(K)
H = HyperellipticCurve(t^11 + 3*t^5 + 5)
H.count_points_hypellfrob()
H.count_points_hypellfrob(n=5)
K = GF(2**7-1)
R.<t> = PolynomialRing(K)
H = HyperellipticCurve(t^13 + 3*t^5 + 5)
H.count_points(n=6)

The base field should be prime:

Sage

sage: K.<z> = GF(19**10)
sage: R.<t> = PolynomialRing(K)
sage: H = HyperellipticCurve(t^9 + (z+1)*t^5 + 1)
sage: H.count_points_hypellfrob()
Traceback (most recent call last):
...
ValueError: hypellfrob does not support non-prime fields

Python

>>> from sage.all import *
>>> K = GF(Integer(19)**Integer(10), names=('z',)); (z,) = K._first_ngens(1)
>>> R = PolynomialRing(K, names=('t',)); (t,) = R._first_ngens(1)
>>> H = HyperellipticCurve(t**Integer(9) + (z+Integer(1))*t**Integer(5) + Integer(1))
>>> H.count_points_hypellfrob()
Traceback (most recent call last):
...
ValueError: hypellfrob does not support non-prime fields

Sage Live

K.<z> = GF(19**10)
R.<t> = PolynomialRing(K)
H = HyperellipticCurve(t^9 + (z+1)*t^5 + 1)
H.count_points_hypellfrob()

and the characteristic should be large enough:

Sage

sage: K = GF(7)
sage: R.<t> = PolynomialRing(K)
sage: H = HyperellipticCurve(t^9 + t^3 + 1)
sage: H.count_points_hypellfrob()
Traceback (most recent call last):
...
ValueError: p=7 should be greater than (2*g+1)(2*N-1)=27

Python

>>> from sage.all import *
>>> K = GF(Integer(7))
>>> R = PolynomialRing(K, names=('t',)); (t,) = R._first_ngens(1)
>>> H = HyperellipticCurve(t**Integer(9) + t**Integer(3) + Integer(1))
>>> H.count_points_hypellfrob()
Traceback (most recent call last):
...
ValueError: p=7 should be greater than (2*g+1)(2*N-1)=27

Sage Live

K = GF(7)
R.<t> = PolynomialRing(K)
H = HyperellipticCurve(t^9 + t^3 + 1)
H.count_points_hypellfrob()

count_points_matrix_traces(n=1, M=None, N=None)[source]¶

Count the number of points on the curve over the first \(n\) extensions of the base field by computing traces of powers of the frobenius matrix. This requires less \(p\)-adic precision than computing the charpoly of the matrix when \(n < g\) where \(g\) is the genus of the curve.

EXAMPLES:

Sage

sage: K = GF(49999)
sage: R.<t> = PolynomialRing(K)
sage: H = HyperellipticCurve(t^19 + t + 1)
sage: H.count_points_matrix_traces(3)
[49491, 2500024375, 124992509154249]

Python

>>> from sage.all import *
>>> K = GF(Integer(49999))
>>> R = PolynomialRing(K, names=('t',)); (t,) = R._first_ngens(1)
>>> H = HyperellipticCurve(t**Integer(19) + t + Integer(1))
>>> H.count_points_matrix_traces(Integer(3))
[49491, 2500024375, 124992509154249]

Sage Live

K = GF(49999)
R.<t> = PolynomialRing(K)
H = HyperellipticCurve(t^19 + t + 1)
H.count_points_matrix_traces(3)

frobenius_matrix(N=None, algorithm='hypellfrob')[source]¶

Compute \(p\)-adic frobenius matrix to precision \(p^N\). If \(N\) not supplied, a default value is selected, which is the minimum needed to recover the charpoly unambiguously.

Note

Currently only implemented using hypellfrob, which means it only works over the prime field \(GF(p)\), and requires \(p > (2g+1)(2N-1)\).

EXAMPLES:

Sage

sage: R.<t> = PolynomialRing(GF(37))
sage: H = HyperellipticCurve(t^5 + t + 2)
sage: H.frobenius_matrix()
[1258 + O(37^2)  925 + O(37^2)  132 + O(37^2)  587 + O(37^2)]
[1147 + O(37^2)  814 + O(37^2)  241 + O(37^2) 1011 + O(37^2)]
[1258 + O(37^2) 1184 + O(37^2) 1105 + O(37^2)  482 + O(37^2)]
[1073 + O(37^2)  999 + O(37^2)  772 + O(37^2)  929 + O(37^2)]

Python

>>> from sage.all import *
>>> R = PolynomialRing(GF(Integer(37)), names=('t',)); (t,) = R._first_ngens(1)
>>> H = HyperellipticCurve(t**Integer(5) + t + Integer(2))
>>> H.frobenius_matrix()
[1258 + O(37^2)  925 + O(37^2)  132 + O(37^2)  587 + O(37^2)]
[1147 + O(37^2)  814 + O(37^2)  241 + O(37^2) 1011 + O(37^2)]
[1258 + O(37^2) 1184 + O(37^2) 1105 + O(37^2)  482 + O(37^2)]
[1073 + O(37^2)  999 + O(37^2)  772 + O(37^2)  929 + O(37^2)]

Sage Live

R.<t> = PolynomialRing(GF(37))
H = HyperellipticCurve(t^5 + t + 2)
H.frobenius_matrix()

The hypellfrob program doesn’t support non-prime fields:

Sage

sage: K.<z> = GF(37**3)
sage: R.<t> = PolynomialRing(K)
sage: H = HyperellipticCurve(t^9 + z*t^3 + 1)
sage: H.frobenius_matrix(algorithm='hypellfrob')
Traceback (most recent call last):
...
NotImplementedError: Computation of Frobenius matrix only implemented
for hyperelliptic curves defined over prime fields.

Python

>>> from sage.all import *
>>> K = GF(Integer(37)**Integer(3), names=('z',)); (z,) = K._first_ngens(1)
>>> R = PolynomialRing(K, names=('t',)); (t,) = R._first_ngens(1)
>>> H = HyperellipticCurve(t**Integer(9) + z*t**Integer(3) + Integer(1))
>>> H.frobenius_matrix(algorithm='hypellfrob')
Traceback (most recent call last):
...
NotImplementedError: Computation of Frobenius matrix only implemented
for hyperelliptic curves defined over prime fields.

Sage Live

K.<z> = GF(37**3)
R.<t> = PolynomialRing(K)
H = HyperellipticCurve(t^9 + z*t^3 + 1)
H.frobenius_matrix(algorithm='hypellfrob')

nor too small characteristic:

Sage

sage: K = GF(7)
sage: R.<t> = PolynomialRing(K)
sage: H = HyperellipticCurve(t^9 + t^3 + 1)
sage: H.frobenius_matrix(algorithm='hypellfrob')
Traceback (most recent call last):
...
ValueError: In the current implementation, p must be greater than (2g+1)(2N-1) = 81

Python

>>> from sage.all import *
>>> K = GF(Integer(7))
>>> R = PolynomialRing(K, names=('t',)); (t,) = R._first_ngens(1)
>>> H = HyperellipticCurve(t**Integer(9) + t**Integer(3) + Integer(1))
>>> H.frobenius_matrix(algorithm='hypellfrob')
Traceback (most recent call last):
...
ValueError: In the current implementation, p must be greater than (2g+1)(2N-1) = 81

Sage Live

K = GF(7)
R.<t> = PolynomialRing(K)
H = HyperellipticCurve(t^9 + t^3 + 1)
H.frobenius_matrix(algorithm='hypellfrob')

frobenius_matrix_hypellfrob(N=None)[source]¶

Compute \(p\)-adic frobenius matrix to precision \(p^N\). If \(N\) not supplied, a default value is selected, which is the minimum needed to recover the charpoly unambiguously.

Note

Implemented using hypellfrob, which means it only works over the prime field \(GF(p)\), and requires \(p > (2g+1)(2N-1)\).

EXAMPLES:

Sage

sage: R.<t> = PolynomialRing(GF(37))
sage: H = HyperellipticCurve(t^5 + t + 2)
sage: H.frobenius_matrix_hypellfrob()
[1258 + O(37^2)  925 + O(37^2)  132 + O(37^2)  587 + O(37^2)]
[1147 + O(37^2)  814 + O(37^2)  241 + O(37^2) 1011 + O(37^2)]
[1258 + O(37^2) 1184 + O(37^2) 1105 + O(37^2)  482 + O(37^2)]
[1073 + O(37^2)  999 + O(37^2)  772 + O(37^2)  929 + O(37^2)]

Python

>>> from sage.all import *
>>> R = PolynomialRing(GF(Integer(37)), names=('t',)); (t,) = R._first_ngens(1)
>>> H = HyperellipticCurve(t**Integer(5) + t + Integer(2))
>>> H.frobenius_matrix_hypellfrob()
[1258 + O(37^2)  925 + O(37^2)  132 + O(37^2)  587 + O(37^2)]
[1147 + O(37^2)  814 + O(37^2)  241 + O(37^2) 1011 + O(37^2)]
[1258 + O(37^2) 1184 + O(37^2) 1105 + O(37^2)  482 + O(37^2)]
[1073 + O(37^2)  999 + O(37^2)  772 + O(37^2)  929 + O(37^2)]

Sage Live

R.<t> = PolynomialRing(GF(37))
H = HyperellipticCurve(t^5 + t + 2)
H.frobenius_matrix_hypellfrob()

The hypellfrob program doesn’t support non-prime fields:

Sage

sage: K.<z> = GF(37**3)
sage: R.<t> = PolynomialRing(K)
sage: H = HyperellipticCurve(t^9 + z*t^3 + 1)
sage: H.frobenius_matrix_hypellfrob()
Traceback (most recent call last):
...
NotImplementedError: Computation of Frobenius matrix only implemented
for hyperelliptic curves defined over prime fields.

Python

>>> from sage.all import *
>>> K = GF(Integer(37)**Integer(3), names=('z',)); (z,) = K._first_ngens(1)
>>> R = PolynomialRing(K, names=('t',)); (t,) = R._first_ngens(1)
>>> H = HyperellipticCurve(t**Integer(9) + z*t**Integer(3) + Integer(1))
>>> H.frobenius_matrix_hypellfrob()
Traceback (most recent call last):
...
NotImplementedError: Computation of Frobenius matrix only implemented
for hyperelliptic curves defined over prime fields.

Sage Live

K.<z> = GF(37**3)
R.<t> = PolynomialRing(K)
H = HyperellipticCurve(t^9 + z*t^3 + 1)
H.frobenius_matrix_hypellfrob()

nor too small characteristic:

Sage

sage: K = GF(7)
sage: R.<t> = PolynomialRing(K)
sage: H = HyperellipticCurve(t^9 + t^3 + 1)
sage: H.frobenius_matrix_hypellfrob()
Traceback (most recent call last):
...
ValueError: In the current implementation, p must be greater than (2g+1)(2N-1) = 81

Python

>>> from sage.all import *
>>> K = GF(Integer(7))
>>> R = PolynomialRing(K, names=('t',)); (t,) = R._first_ngens(1)
>>> H = HyperellipticCurve(t**Integer(9) + t**Integer(3) + Integer(1))
>>> H.frobenius_matrix_hypellfrob()
Traceback (most recent call last):
...
ValueError: In the current implementation, p must be greater than (2g+1)(2N-1) = 81

Sage Live

K = GF(7)
R.<t> = PolynomialRing(K)
H = HyperellipticCurve(t^9 + t^3 + 1)
H.frobenius_matrix_hypellfrob()

frobenius_polynomial()[source]¶

Compute the charpoly of frobenius, as an element of \(\ZZ[x]\).

EXAMPLES:

Sage

sage: R.<t> = PolynomialRing(GF(37))
sage: H = HyperellipticCurve(t^5 + t + 2)
sage: H.frobenius_polynomial()
x^4 + x^3 - 52*x^2 + 37*x + 1369

Python

>>> from sage.all import *
>>> R = PolynomialRing(GF(Integer(37)), names=('t',)); (t,) = R._first_ngens(1)
>>> H = HyperellipticCurve(t**Integer(5) + t + Integer(2))
>>> H.frobenius_polynomial()
x^4 + x^3 - 52*x^2 + 37*x + 1369

Sage Live

R.<t> = PolynomialRing(GF(37))
H = HyperellipticCurve(t^5 + t + 2)
H.frobenius_polynomial()

A quadratic twist:

Sage

sage: H = HyperellipticCurve(2*t^5 + 2*t + 4)
sage: H.frobenius_polynomial()
x^4 - x^3 - 52*x^2 - 37*x + 1369

Python

>>> from sage.all import *
>>> H = HyperellipticCurve(Integer(2)*t**Integer(5) + Integer(2)*t + Integer(4))
>>> H.frobenius_polynomial()
x^4 - x^3 - 52*x^2 - 37*x + 1369

Sage Live

H = HyperellipticCurve(2*t^5 + 2*t + 4)
H.frobenius_polynomial()

Slightly larger example:

Sage

sage: K = GF(2003)
sage: R.<t> = PolynomialRing(K)
sage: H = HyperellipticCurve(t^7 + 487*t^5 + 9*t + 1)
sage: H.frobenius_polynomial()
x^6 - 14*x^5 + 1512*x^4 - 66290*x^3 + 3028536*x^2 - 56168126*x + 8036054027

Python

>>> from sage.all import *
>>> K = GF(Integer(2003))
>>> R = PolynomialRing(K, names=('t',)); (t,) = R._first_ngens(1)
>>> H = HyperellipticCurve(t**Integer(7) + Integer(487)*t**Integer(5) + Integer(9)*t + Integer(1))
>>> H.frobenius_polynomial()
x^6 - 14*x^5 + 1512*x^4 - 66290*x^3 + 3028536*x^2 - 56168126*x + 8036054027

Sage Live

K = GF(2003)
R.<t> = PolynomialRing(K)
H = HyperellipticCurve(t^7 + 487*t^5 + 9*t + 1)
H.frobenius_polynomial()

Curves defined over a non-prime field of odd characteristic, or an odd prime field which is too small compared to the genus, are supported via PARI:

Sage

sage: K.<z> = GF(23**3)
sage: R.<t> = PolynomialRing(K)
sage: H = HyperellipticCurve(t^3 + z*t + 4)
sage: H.frobenius_polynomial()
x^2 - 15*x + 12167

sage: K.<z> = GF(3**3)
sage: R.<t> = PolynomialRing(K)
sage: H = HyperellipticCurve(t^5 + z*t + z**3)
sage: H.frobenius_polynomial()
x^4 - 3*x^3 + 10*x^2 - 81*x + 729

Python

>>> from sage.all import *
>>> K = GF(Integer(23)**Integer(3), names=('z',)); (z,) = K._first_ngens(1)
>>> R = PolynomialRing(K, names=('t',)); (t,) = R._first_ngens(1)
>>> H = HyperellipticCurve(t**Integer(3) + z*t + Integer(4))
>>> H.frobenius_polynomial()
x^2 - 15*x + 12167

>>> K = GF(Integer(3)**Integer(3), names=('z',)); (z,) = K._first_ngens(1)
>>> R = PolynomialRing(K, names=('t',)); (t,) = R._first_ngens(1)
>>> H = HyperellipticCurve(t**Integer(5) + z*t + z**Integer(3))
>>> H.frobenius_polynomial()
x^4 - 3*x^3 + 10*x^2 - 81*x + 729

Sage Live

K.<z> = GF(23**3)
R.<t> = PolynomialRing(K)
H = HyperellipticCurve(t^3 + z*t + 4)
H.frobenius_polynomial()
K.<z> = GF(3**3)
R.<t> = PolynomialRing(K)
H = HyperellipticCurve(t^5 + z*t + z**3)
H.frobenius_polynomial()

Over prime fields of odd characteristic, \(h\) may be nonzero:

Sage

sage: K = GF(101)
sage: R.<t> = PolynomialRing(K)
sage: H = HyperellipticCurve(t^5 + 27*t + 3, t)
sage: H.frobenius_polynomial()
x^4 + 2*x^3 - 58*x^2 + 202*x + 10201

Python

>>> from sage.all import *
>>> K = GF(Integer(101))
>>> R = PolynomialRing(K, names=('t',)); (t,) = R._first_ngens(1)
>>> H = HyperellipticCurve(t**Integer(5) + Integer(27)*t + Integer(3), t)
>>> H.frobenius_polynomial()
x^4 + 2*x^3 - 58*x^2 + 202*x + 10201

Sage Live

K = GF(101)
R.<t> = PolynomialRing(K)
H = HyperellipticCurve(t^5 + 27*t + 3, t)
H.frobenius_polynomial()

Over prime fields of odd characteristic, \(f\) may have even degree:

Sage

sage: H = HyperellipticCurve(t^6 + 27*t + 3)
sage: H.frobenius_polynomial()
x^4 + 25*x^3 + 322*x^2 + 2525*x + 10201

Python

>>> from sage.all import *
>>> H = HyperellipticCurve(t**Integer(6) + Integer(27)*t + Integer(3))
>>> H.frobenius_polynomial()
x^4 + 25*x^3 + 322*x^2 + 2525*x + 10201

Sage Live

H = HyperellipticCurve(t^6 + 27*t + 3)
H.frobenius_polynomial()

In even characteristic, the naive algorithm could cover all cases because we can easily check for squareness in quotient rings of polynomial rings over finite fields but these rings unfortunately do not support iteration:

Sage

sage: K.<z> = GF(2**5)
sage: R.<t> = PolynomialRing(K)
sage: H = HyperellipticCurve(t^5 + z*t + z**3, t)
sage: H.frobenius_polynomial()
x^4 - x^3 + 16*x^2 - 32*x + 1024

Python

>>> from sage.all import *
>>> K = GF(Integer(2)**Integer(5), names=('z',)); (z,) = K._first_ngens(1)
>>> R = PolynomialRing(K, names=('t',)); (t,) = R._first_ngens(1)
>>> H = HyperellipticCurve(t**Integer(5) + z*t + z**Integer(3), t)
>>> H.frobenius_polynomial()
x^4 - x^3 + 16*x^2 - 32*x + 1024

Sage Live

K.<z> = GF(2**5)
R.<t> = PolynomialRing(K)
H = HyperellipticCurve(t^5 + z*t + z**3, t)
H.frobenius_polynomial()

frobenius_polynomial_cardinalities(a=None)[source]¶

Compute the charpoly of frobenius, as an element of \(\ZZ[x]\), by computing the number of points on the curve over \(g\) extensions of the base field where \(g\) is the genus of the curve.

Warning

This is highly inefficient when the base field or the genus of the curve are large.

EXAMPLES:

Sage

sage: R.<t> = PolynomialRing(GF(37))
sage: H = HyperellipticCurve(t^5 + t + 2)
sage: H.frobenius_polynomial_cardinalities()
x^4 + x^3 - 52*x^2 + 37*x + 1369

Python

>>> from sage.all import *
>>> R = PolynomialRing(GF(Integer(37)), names=('t',)); (t,) = R._first_ngens(1)
>>> H = HyperellipticCurve(t**Integer(5) + t + Integer(2))
>>> H.frobenius_polynomial_cardinalities()
x^4 + x^3 - 52*x^2 + 37*x + 1369

Sage Live

R.<t> = PolynomialRing(GF(37))
H = HyperellipticCurve(t^5 + t + 2)
H.frobenius_polynomial_cardinalities()

A quadratic twist:

Sage

sage: H = HyperellipticCurve(2*t^5 + 2*t + 4)
sage: H.frobenius_polynomial_cardinalities()
x^4 - x^3 - 52*x^2 - 37*x + 1369

Python

>>> from sage.all import *
>>> H = HyperellipticCurve(Integer(2)*t**Integer(5) + Integer(2)*t + Integer(4))
>>> H.frobenius_polynomial_cardinalities()
x^4 - x^3 - 52*x^2 - 37*x + 1369

Sage Live

H = HyperellipticCurve(2*t^5 + 2*t + 4)
H.frobenius_polynomial_cardinalities()

Curve over a non-prime field:

Sage

sage: K.<z> = GF(7**2)
sage: R.<t> = PolynomialRing(K)
sage: H = HyperellipticCurve(t^5 + z*t + z^2)
sage: H.frobenius_polynomial_cardinalities()
x^4 + 8*x^3 + 70*x^2 + 392*x + 2401

Python

>>> from sage.all import *
>>> K = GF(Integer(7)**Integer(2), names=('z',)); (z,) = K._first_ngens(1)
>>> R = PolynomialRing(K, names=('t',)); (t,) = R._first_ngens(1)
>>> H = HyperellipticCurve(t**Integer(5) + z*t + z**Integer(2))
>>> H.frobenius_polynomial_cardinalities()
x^4 + 8*x^3 + 70*x^2 + 392*x + 2401

Sage Live

K.<z> = GF(7**2)
R.<t> = PolynomialRing(K)
H = HyperellipticCurve(t^5 + z*t + z^2)
H.frobenius_polynomial_cardinalities()

This method may actually be useful when hypellfrob does not work:

Sage

sage: K = GF(7)
sage: R.<t> = PolynomialRing(K)
sage: H = HyperellipticCurve(t^9 + t^3 + 1)
sage: H.frobenius_polynomial_matrix(algorithm='hypellfrob')
Traceback (most recent call last):
...
ValueError: In the current implementation, p must be greater than (2g+1)(2N-1) = 81
sage: H.frobenius_polynomial_cardinalities()
x^8 - 5*x^7 + 7*x^6 + 36*x^5 - 180*x^4 + 252*x^3 + 343*x^2 - 1715*x + 2401

Python

>>> from sage.all import *
>>> K = GF(Integer(7))
>>> R = PolynomialRing(K, names=('t',)); (t,) = R._first_ngens(1)
>>> H = HyperellipticCurve(t**Integer(9) + t**Integer(3) + Integer(1))
>>> H.frobenius_polynomial_matrix(algorithm='hypellfrob')
Traceback (most recent call last):
...
ValueError: In the current implementation, p must be greater than (2g+1)(2N-1) = 81
>>> H.frobenius_polynomial_cardinalities()
x^8 - 5*x^7 + 7*x^6 + 36*x^5 - 180*x^4 + 252*x^3 + 343*x^2 - 1715*x + 2401

Sage Live

K = GF(7)
R.<t> = PolynomialRing(K)
H = HyperellipticCurve(t^9 + t^3 + 1)
H.frobenius_polynomial_matrix(algorithm='hypellfrob')
H.frobenius_polynomial_cardinalities()

frobenius_polynomial_matrix(M=None, algorithm='hypellfrob')[source]¶

Compute the charpoly of frobenius, as an element of \(\ZZ[x]\), by computing the charpoly of the frobenius matrix.

This is currently only supported when the base field is prime and large enough using the hypellfrob library.

EXAMPLES:

Sage

sage: R.<t> = PolynomialRing(GF(37))
sage: H = HyperellipticCurve(t^5 + t + 2)
sage: H.frobenius_polynomial_matrix()
x^4 + x^3 - 52*x^2 + 37*x + 1369

Python

>>> from sage.all import *
>>> R = PolynomialRing(GF(Integer(37)), names=('t',)); (t,) = R._first_ngens(1)
>>> H = HyperellipticCurve(t**Integer(5) + t + Integer(2))
>>> H.frobenius_polynomial_matrix()
x^4 + x^3 - 52*x^2 + 37*x + 1369

Sage Live

R.<t> = PolynomialRing(GF(37))
H = HyperellipticCurve(t^5 + t + 2)
H.frobenius_polynomial_matrix()

A quadratic twist:

Sage

sage: H = HyperellipticCurve(2*t^5 + 2*t + 4)
sage: H.frobenius_polynomial_matrix()
x^4 - x^3 - 52*x^2 - 37*x + 1369

Python

>>> from sage.all import *
>>> H = HyperellipticCurve(Integer(2)*t**Integer(5) + Integer(2)*t + Integer(4))
>>> H.frobenius_polynomial_matrix()
x^4 - x^3 - 52*x^2 - 37*x + 1369

Sage Live

H = HyperellipticCurve(2*t^5 + 2*t + 4)
H.frobenius_polynomial_matrix()

Curves defined over larger prime fields:

Sage

sage: K = GF(49999)
sage: R.<t> = PolynomialRing(K)
sage: H = HyperellipticCurve(t^9 + t^5 + 1)
sage: H.frobenius_polynomial_matrix()
x^8 + 281*x^7 + 55939*x^6 + 14144175*x^5 + 3156455369*x^4 + 707194605825*x^3
+ 139841906155939*x^2 + 35122892542149719*x + 6249500014999800001
sage: H = HyperellipticCurve(t^15 + t^5 + 1)
sage: H.frobenius_polynomial_matrix()  # long time, 8s on a Corei7
x^14 - 76*x^13 + 220846*x^12 - 12984372*x^11 + 24374326657*x^10 - 1203243210304*x^9
+ 1770558798515792*x^8 - 74401511415210496*x^7 + 88526169366991084208*x^6
- 3007987702642212810304*x^5 + 3046608028331197124223343*x^4
- 81145833008762983138584372*x^3 + 69007473838551978905211279154*x^2
- 1187357507124810002849977200076*x + 781140631562281254374947500349999

Python

>>> from sage.all import *
>>> K = GF(Integer(49999))
>>> R = PolynomialRing(K, names=('t',)); (t,) = R._first_ngens(1)
>>> H = HyperellipticCurve(t**Integer(9) + t**Integer(5) + Integer(1))
>>> H.frobenius_polynomial_matrix()
x^8 + 281*x^7 + 55939*x^6 + 14144175*x^5 + 3156455369*x^4 + 707194605825*x^3
+ 139841906155939*x^2 + 35122892542149719*x + 6249500014999800001
>>> H = HyperellipticCurve(t**Integer(15) + t**Integer(5) + Integer(1))
>>> H.frobenius_polynomial_matrix()  # long time, 8s on a Corei7
x^14 - 76*x^13 + 220846*x^12 - 12984372*x^11 + 24374326657*x^10 - 1203243210304*x^9
+ 1770558798515792*x^8 - 74401511415210496*x^7 + 88526169366991084208*x^6
- 3007987702642212810304*x^5 + 3046608028331197124223343*x^4
- 81145833008762983138584372*x^3 + 69007473838551978905211279154*x^2
- 1187357507124810002849977200076*x + 781140631562281254374947500349999

Sage Live

K = GF(49999)
R.<t> = PolynomialRing(K)
H = HyperellipticCurve(t^9 + t^5 + 1)
H.frobenius_polynomial_matrix()
H = HyperellipticCurve(t^15 + t^5 + 1)
H.frobenius_polynomial_matrix()  # long time, 8s on a Corei7

This hypellfrob program doesn’t support non-prime fields:

Sage

sage: K.<z> = GF(37**3)
sage: R.<t> = PolynomialRing(K)
sage: H = HyperellipticCurve(t^9 + z*t^3 + 1)
sage: H.frobenius_polynomial_matrix(algorithm='hypellfrob')
Traceback (most recent call last):
...
NotImplementedError: Computation of Frobenius matrix only implemented
for hyperelliptic curves defined over prime fields.

Python

>>> from sage.all import *
>>> K = GF(Integer(37)**Integer(3), names=('z',)); (z,) = K._first_ngens(1)
>>> R = PolynomialRing(K, names=('t',)); (t,) = R._first_ngens(1)
>>> H = HyperellipticCurve(t**Integer(9) + z*t**Integer(3) + Integer(1))
>>> H.frobenius_polynomial_matrix(algorithm='hypellfrob')
Traceback (most recent call last):
...
NotImplementedError: Computation of Frobenius matrix only implemented
for hyperelliptic curves defined over prime fields.

Sage Live

K.<z> = GF(37**3)
R.<t> = PolynomialRing(K)
H = HyperellipticCurve(t^9 + z*t^3 + 1)
H.frobenius_polynomial_matrix(algorithm='hypellfrob')

frobenius_polynomial_pari()[source]¶

Compute the charpoly of frobenius, as an element of \(\ZZ[x]\), by calling the PARI function hyperellcharpoly.

EXAMPLES:

Sage

sage: R.<t> = PolynomialRing(GF(37))
sage: H = HyperellipticCurve(t^5 + t + 2)
sage: H.frobenius_polynomial_pari()
x^4 + x^3 - 52*x^2 + 37*x + 1369

Python

>>> from sage.all import *
>>> R = PolynomialRing(GF(Integer(37)), names=('t',)); (t,) = R._first_ngens(1)
>>> H = HyperellipticCurve(t**Integer(5) + t + Integer(2))
>>> H.frobenius_polynomial_pari()
x^4 + x^3 - 52*x^2 + 37*x + 1369

Sage Live

R.<t> = PolynomialRing(GF(37))
H = HyperellipticCurve(t^5 + t + 2)
H.frobenius_polynomial_pari()

A quadratic twist:

Sage

sage: H = HyperellipticCurve(2*t^5 + 2*t + 4)
sage: H.frobenius_polynomial_pari()
x^4 - x^3 - 52*x^2 - 37*x + 1369

Python

>>> from sage.all import *
>>> H = HyperellipticCurve(Integer(2)*t**Integer(5) + Integer(2)*t + Integer(4))
>>> H.frobenius_polynomial_pari()
x^4 - x^3 - 52*x^2 - 37*x + 1369

Sage Live

H = HyperellipticCurve(2*t^5 + 2*t + 4)
H.frobenius_polynomial_pari()

Slightly larger example:

Sage

sage: K = GF(2003)
sage: R.<t> = PolynomialRing(K)
sage: H = HyperellipticCurve(t^7 + 487*t^5 + 9*t + 1)
sage: H.frobenius_polynomial_pari()
x^6 - 14*x^5 + 1512*x^4 - 66290*x^3 + 3028536*x^2 - 56168126*x + 8036054027

Python

>>> from sage.all import *
>>> K = GF(Integer(2003))
>>> R = PolynomialRing(K, names=('t',)); (t,) = R._first_ngens(1)
>>> H = HyperellipticCurve(t**Integer(7) + Integer(487)*t**Integer(5) + Integer(9)*t + Integer(1))
>>> H.frobenius_polynomial_pari()
x^6 - 14*x^5 + 1512*x^4 - 66290*x^3 + 3028536*x^2 - 56168126*x + 8036054027

Sage Live

K = GF(2003)
R.<t> = PolynomialRing(K)
H = HyperellipticCurve(t^7 + 487*t^5 + 9*t + 1)
H.frobenius_polynomial_pari()

Curves defined over a non-prime field are supported as well:

Sage

sage: K.<a> = GF(7^2)
sage: R.<t> = PolynomialRing(K)
sage: H = HyperellipticCurve(t^5 + a*t + 1)
sage: H.frobenius_polynomial_pari()
x^4 + 4*x^3 + 84*x^2 + 196*x + 2401

sage: K.<z> = GF(23**3)
sage: R.<t> = PolynomialRing(K)
sage: H = HyperellipticCurve(t^3 + z*t + 4)
sage: H.frobenius_polynomial_pari()
x^2 - 15*x + 12167

Python

>>> from sage.all import *
>>> K = GF(Integer(7)**Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> R = PolynomialRing(K, names=('t',)); (t,) = R._first_ngens(1)
>>> H = HyperellipticCurve(t**Integer(5) + a*t + Integer(1))
>>> H.frobenius_polynomial_pari()
x^4 + 4*x^3 + 84*x^2 + 196*x + 2401

>>> K = GF(Integer(23)**Integer(3), names=('z',)); (z,) = K._first_ngens(1)
>>> R = PolynomialRing(K, names=('t',)); (t,) = R._first_ngens(1)
>>> H = HyperellipticCurve(t**Integer(3) + z*t + Integer(4))
>>> H.frobenius_polynomial_pari()
x^2 - 15*x + 12167

Sage Live

K.<a> = GF(7^2)
R.<t> = PolynomialRing(K)
H = HyperellipticCurve(t^5 + a*t + 1)
H.frobenius_polynomial_pari()
K.<z> = GF(23**3)
R.<t> = PolynomialRing(K)
H = HyperellipticCurve(t^3 + z*t + 4)
H.frobenius_polynomial_pari()

Over prime fields of odd characteristic, \(h\) may be nonzero:

Sage

sage: K = GF(101)
sage: R.<t> = PolynomialRing(K)
sage: H = HyperellipticCurve(t^5 + 27*t + 3, t)
sage: H.frobenius_polynomial_pari()
x^4 + 2*x^3 - 58*x^2 + 202*x + 10201

Python

>>> from sage.all import *
>>> K = GF(Integer(101))
>>> R = PolynomialRing(K, names=('t',)); (t,) = R._first_ngens(1)
>>> H = HyperellipticCurve(t**Integer(5) + Integer(27)*t + Integer(3), t)
>>> H.frobenius_polynomial_pari()
x^4 + 2*x^3 - 58*x^2 + 202*x + 10201

Sage Live

K = GF(101)
R.<t> = PolynomialRing(K)
H = HyperellipticCurve(t^5 + 27*t + 3, t)
H.frobenius_polynomial_pari()

p_rank()[source]¶

INPUT:

self – Hyperelliptic Curve of the form \(y^2 = f(x)\) over a finite field, \(\GF{q}\)

OUTPUT: p-rank

EXAMPLES:

Sage

sage: K.<x> = GF(49, 'x')[]
sage: C = HyperellipticCurve(x^5 + 1, 0)
sage: C.p_rank()
0

sage: K.<x> = GF(9, 'x')[]
sage: C = HyperellipticCurve(x^7 - 1, 0)
sage: C.p_rank()
0

sage: P.<x> = GF(9, 'a')[]
sage: E = HyperellipticCurve(x^29 + 1, 0)
sage: E.p_rank()
0

Python

>>> from sage.all import *
>>> K = GF(Integer(49), 'x')['x']; (x,) = K._first_ngens(1)
>>> C = HyperellipticCurve(x**Integer(5) + Integer(1), Integer(0))
>>> C.p_rank()
0

>>> K = GF(Integer(9), 'x')['x']; (x,) = K._first_ngens(1)
>>> C = HyperellipticCurve(x**Integer(7) - Integer(1), Integer(0))
>>> C.p_rank()
0

>>> P = GF(Integer(9), 'a')['x']; (x,) = P._first_ngens(1)
>>> E = HyperellipticCurve(x**Integer(29) + Integer(1), Integer(0))
>>> E.p_rank()
0

Sage Live

K.<x> = GF(49, 'x')[]
C = HyperellipticCurve(x^5 + 1, 0)
C.p_rank()
K.<x> = GF(9, 'x')[]
C = HyperellipticCurve(x^7 - 1, 0)
C.p_rank()
P.<x> = GF(9, 'a')[]
E = HyperellipticCurve(x^29 + 1, 0)
E.p_rank()

points()[source]¶

All the points on this hyperelliptic curve.

EXAMPLES:

Sage

sage: x = polygen(GF(7))
sage: C = HyperellipticCurve(x^7 - x^2 - 1)
sage: C.points()
[(0 : 1 : 0), (2 : 5 : 1), (2 : 2 : 1), (3 : 0 : 1), (4 : 6 : 1),
 (4 : 1 : 1), (5 : 0 : 1), (6 : 5 : 1), (6 : 2 : 1)]

Python

>>> from sage.all import *
>>> x = polygen(GF(Integer(7)))
>>> C = HyperellipticCurve(x**Integer(7) - x**Integer(2) - Integer(1))
>>> C.points()
[(0 : 1 : 0), (2 : 5 : 1), (2 : 2 : 1), (3 : 0 : 1), (4 : 6 : 1),
 (4 : 1 : 1), (5 : 0 : 1), (6 : 5 : 1), (6 : 2 : 1)]

Sage Live

x = polygen(GF(7))
C = HyperellipticCurve(x^7 - x^2 - 1)
C.points()

Sage

sage: x = polygen(GF(121, 'a'))
sage: C = HyperellipticCurve(x^5 + x - 1, x^2 + 2)
sage: len(C.points())
122

Python

>>> from sage.all import *
>>> x = polygen(GF(Integer(121), 'a'))
>>> C = HyperellipticCurve(x**Integer(5) + x - Integer(1), x**Integer(2) + Integer(2))
>>> len(C.points())
122

Sage Live

x = polygen(GF(121, 'a'))
C = HyperellipticCurve(x^5 + x - 1, x^2 + 2)
len(C.points())

Conics are allowed (the issue reported at Issue #11800 has been resolved):

Sage

sage: R.<x> = GF(7)[]
sage: H = HyperellipticCurve(3*x^2 + 5*x + 1)
sage: H.points()
[(0 : 6 : 1), (0 : 1 : 1), (1 : 4 : 1), (1 : 3 : 1), (2 : 4 : 1),
 (2 : 3 : 1), (3 : 6 : 1), (3 : 1 : 1)]

Python

>>> from sage.all import *
>>> R = GF(Integer(7))['x']; (x,) = R._first_ngens(1)
>>> H = HyperellipticCurve(Integer(3)*x**Integer(2) + Integer(5)*x + Integer(1))
>>> H.points()
[(0 : 6 : 1), (0 : 1 : 1), (1 : 4 : 1), (1 : 3 : 1), (2 : 4 : 1),
 (2 : 3 : 1), (3 : 6 : 1), (3 : 1 : 1)]

Sage Live

R.<x> = GF(7)[]
H = HyperellipticCurve(3*x^2 + 5*x + 1)
H.points()

The method currently lists points on the plane projective model, that is the closure in \(\mathbb{P}^2\) of the curve defined by \(y^2+hy=f\). This means that one point \((0:1:0)\) at infinity is returned if the degree of the curve is at least 4 and \(\deg(f)>\deg(h)+1\). This point is a singular point of the plane model. Later implementations may consider a smooth model instead since that would be a more relevant object. Then, for a curve whose only singularity is at \((0:1:0)\), the point at infinity would be replaced by a number of rational points of the smooth model. We illustrate this with an example of a genus 2 hyperelliptic curve:

Sage

sage: R.<x>=GF(11)[]
sage: H = HyperellipticCurve(x*(x+1)*(x+2)*(x+3)*(x+4)*(x+5))
sage: H.points()
[(0 : 1 : 0), (0 : 0 : 1), (1 : 7 : 1), (1 : 4 : 1), (5 : 7 : 1), (5 : 4 : 1),
 (6 : 0 : 1), (7 : 0 : 1), (8 : 0 : 1), (9 : 0 : 1), (10 : 0 : 1)]

Python

>>> from sage.all import *
>>> R = GF(Integer(11))['x']; (x,) = R._first_ngens(1)
>>> H = HyperellipticCurve(x*(x+Integer(1))*(x+Integer(2))*(x+Integer(3))*(x+Integer(4))*(x+Integer(5)))
>>> H.points()
[(0 : 1 : 0), (0 : 0 : 1), (1 : 7 : 1), (1 : 4 : 1), (5 : 7 : 1), (5 : 4 : 1),
 (6 : 0 : 1), (7 : 0 : 1), (8 : 0 : 1), (9 : 0 : 1), (10 : 0 : 1)]

Sage Live

R.<x>=GF(11)[]
H = HyperellipticCurve(x*(x+1)*(x+2)*(x+3)*(x+4)*(x+5))
H.points()

The plane model of the genus 2 hyperelliptic curve in the above example is the curve in \(\mathbb{P}^2\) defined by \(y^2z^4=g(x,z)\) where \(g(x,z)=x(x+z)(x+2z)(x+3z)(x+4z)(x+5z).\) This model has one point at infinity \((0:1:0)\) which is also the only singular point of the plane model. In contrast, the hyperelliptic curve is smooth and imbeds via the equation \(y^2=g(x,z)\) into weighted projected space \(\mathbb{P}(1,3,1)\). The latter model has two points at infinity: \((1:1:0)\) and \((1:-1:0)\).

zeta_function()[source]¶

Compute the zeta function of the hyperelliptic curve.

EXAMPLES:

Sage

sage: F = GF(2); R.<t> = F[]
sage: H = HyperellipticCurve(t^9 + t, t^4)
sage: H.zeta_function()
(16*x^8 + 8*x^7 + 8*x^6 + 4*x^5 + 6*x^4 + 2*x^3 + 2*x^2 + x + 1)/(2*x^2 - 3*x + 1)

sage: F.<a> = GF(4); R.<t> = F[]
sage: H = HyperellipticCurve(t^5 + t^3 + t^2 + t + 1, t^2 + t + 1)
sage: H.zeta_function()
(16*x^4 + 8*x^3 + x^2 + 2*x + 1)/(4*x^2 - 5*x + 1)

sage: F.<a> = GF(9); R.<t> = F[]
sage: H = HyperellipticCurve(t^5 + a*t)
sage: H.zeta_function()
(81*x^4 + 72*x^3 + 32*x^2 + 8*x + 1)/(9*x^2 - 10*x + 1)

sage: R.<t> = PolynomialRing(GF(37))
sage: H = HyperellipticCurve(t^5 + t + 2)
sage: H.zeta_function()
(1369*x^4 + 37*x^3 - 52*x^2 + x + 1)/(37*x^2 - 38*x + 1)

Python

>>> from sage.all import *
>>> F = GF(Integer(2)); R = F['t']; (t,) = R._first_ngens(1)
>>> H = HyperellipticCurve(t**Integer(9) + t, t**Integer(4))
>>> H.zeta_function()
(16*x^8 + 8*x^7 + 8*x^6 + 4*x^5 + 6*x^4 + 2*x^3 + 2*x^2 + x + 1)/(2*x^2 - 3*x + 1)

>>> F = GF(Integer(4), names=('a',)); (a,) = F._first_ngens(1); R = F['t']; (t,) = R._first_ngens(1)
>>> H = HyperellipticCurve(t**Integer(5) + t**Integer(3) + t**Integer(2) + t + Integer(1), t**Integer(2) + t + Integer(1))
>>> H.zeta_function()
(16*x^4 + 8*x^3 + x^2 + 2*x + 1)/(4*x^2 - 5*x + 1)

>>> F = GF(Integer(9), names=('a',)); (a,) = F._first_ngens(1); R = F['t']; (t,) = R._first_ngens(1)
>>> H = HyperellipticCurve(t**Integer(5) + a*t)
>>> H.zeta_function()
(81*x^4 + 72*x^3 + 32*x^2 + 8*x + 1)/(9*x^2 - 10*x + 1)

>>> R = PolynomialRing(GF(Integer(37)), names=('t',)); (t,) = R._first_ngens(1)
>>> H = HyperellipticCurve(t**Integer(5) + t + Integer(2))
>>> H.zeta_function()
(1369*x^4 + 37*x^3 - 52*x^2 + x + 1)/(37*x^2 - 38*x + 1)

Sage Live

F = GF(2); R.<t> = F[]
H = HyperellipticCurve(t^9 + t, t^4)
H.zeta_function()
F.<a> = GF(4); R.<t> = F[]
H = HyperellipticCurve(t^5 + t^3 + t^2 + t + 1, t^2 + t + 1)
H.zeta_function()
F.<a> = GF(9); R.<t> = F[]
H = HyperellipticCurve(t^5 + a*t)
H.zeta_function()
R.<t> = PolynomialRing(GF(37))
H = HyperellipticCurve(t^5 + t + 2)
H.zeta_function()

A quadratic twist:

Sage

sage: R.<t> = PolynomialRing(GF(37))
sage: H = HyperellipticCurve(2*t^5 + 2*t + 4)
sage: H.zeta_function()
(1369*x^4 - 37*x^3 - 52*x^2 - x + 1)/(37*x^2 - 38*x + 1)

Python

>>> from sage.all import *
>>> R = PolynomialRing(GF(Integer(37)), names=('t',)); (t,) = R._first_ngens(1)
>>> H = HyperellipticCurve(Integer(2)*t**Integer(5) + Integer(2)*t + Integer(4))
>>> H.zeta_function()
(1369*x^4 - 37*x^3 - 52*x^2 - x + 1)/(37*x^2 - 38*x + 1)

Sage Live

R.<t> = PolynomialRing(GF(37))
H = HyperellipticCurve(2*t^5 + 2*t + 4)
H.zeta_function()