Hyperelliptic curves over a finite field

EXAMPLES:

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)]

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)

Bases: sage.schemes.hyperelliptic_curves.hyperelliptic_generic.HyperellipticCurve_generic, sage.schemes.curves.projective_curve.ProjectivePlaneCurve_finite_field

Cartier_matrix()

INPUT:

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

OUTPUT:

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

REFERENCES:

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

EXAMPLES:

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]

Hasse_Witt()

INPUT:

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

OUTPUT:

• N : 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: 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]

a_number()

INPUT:

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

OUTPUT:

• a : a-number

EXAMPLES:

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

cardinality(extension_degree=1)

Count points on a single extension of the base field.

EXAMPLES:

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

The following example shows that trac ticket #20391 has been resolved:

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

cardinality_exhaustive(extension_degree=1, algorithm=None)

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

EXAMPLES:

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]

cardinality_hypellfrob(extension_degree=1, algorithm=None)

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

EXAMPLES:

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

count_points(n=1)

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: 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]

This example shows that trac ticket #20391 is resolved:

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

count_points_exhaustive(n=1, naive=False)

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: 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]

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

This behavior can be disabled by passing naive=True:

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

count_points_frobenius_polynomial(n=1, f=None)

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

EXAMPLES:

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

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

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

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

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]

count_points_hypellfrob(n=1, N=None, algorithm=None)

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: 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]

The base field should be prime:

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

and the characteristic should be large enough:

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

count_points_matrix_traces(n=1, M=None, N=None)

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: 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]

frobenius_matrix(N=None, algorithm='hypellfrob')

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: 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)]

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

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.

nor too small characteristic:

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

frobenius_matrix_hypellfrob(N=None)

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: 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)]

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

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.

nor too small characteristic:

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

frobenius_polynomial()

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

EXAMPLES:

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

A quadratic twist:

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

Slightly larger example:

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

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: 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

Over prime fields of odd characteristic, \(h\) may be non-zero:

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

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

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

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: 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

frobenius_polynomial_cardinalities(a=None)

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: 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

A quadratic twist:

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

Curve over a non-prime field:

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

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

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

frobenius_polynomial_matrix(M=None, algorithm='hypellfrob')

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: 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

A quadratic twist:

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

Curves defined over larger prime fields:

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

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

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.

frobenius_polynomial_pari()

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

EXAMPLES:

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

A quadratic twist:

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

Slightly larger example:

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

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

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

Over prime fields of odd characteristic, \(h\) may be non-zero:

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

p_rank()

INPUT:

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

OUTPUT:

• pr :p-rank

EXAMPLES:

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

points()

All the points on this hyperelliptic curve.

EXAMPLES:

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)]

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

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

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)]

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: 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)]

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()

Compute the zeta function of the hyperelliptic curve.

EXAMPLES:

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)

A quadratic twist:

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)