# Hyperelliptic curves over a finite field¶

AUTHORS:

• David Kohel (2006)
• Alyson Deines, Marina Gresham, Gagan Sekhon, (2010)
• Daniel Krenn (2011)
• Jean-Pierre Flori, Jan Tuitman (2013)

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)]

class sage.schemes.hyperelliptic_curves.hyperelliptic_finite_field.HyperellipticCurve_finite_field(PP, f, h=None, names=None, genus=None)
Cartier_matrix()

INPUT:

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

OUTPUT:

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

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.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]


TESTS:

sage: K.<x>=GF(2,'x')[]
sage: C=HyperellipticCurve(x^7-1,x)
sage: C.Cartier_matrix()
Traceback (most recent call last):
...
ValueError: p must be odd

sage: K.<x>=GF(5,'x')[]
sage: C=HyperellipticCurve(x^7-1,4)
sage: C.Cartier_matrix()
Traceback (most recent call last):
...
ValueError: E must be of the form y^2 = f(x)

sage: K.<x>=GF(5,'x')[]
sage: C=HyperellipticCurve(x^8-1,0)
sage: C.Cartier_matrix()
Traceback (most recent call last):
...
ValueError: In this implementation the degree of f must be odd

sage: K.<x>=GF(5,'x')[]
sage: C=HyperellipticCurve(x^5+1,0,check_squarefree=False)
sage: C.Cartier_matrix()
Traceback (most recent call last):
...
ValueError: curve is not smooth

Hasse_Witt()

INPUT:

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

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, $$\mathbb{F}_q$$

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

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 prgoram.

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]

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 only works over the prime field GF(p), and must have 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

Currently only implemented using hypellfrob, which means only works over the prime field GF(p), and must have 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


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 are supported as well, but a naive algorithm is used; especially when g = 1, fast point counting on elliptic curves should be used:

sage: K.<z> = GF(23**3)
sage: R.<t> = PolynomialRing(K)
sage: H = HyperellipticCurve(t^3 + z*t + 4)
sage: H.frobenius_polynomial() # long time, 4s on a Corei7
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, when h is non-zero, this naive algorithm is currently used as well, whereas we should rather use another defining equation:

sage: K = GF(101)
sage: R.<t> = PolynomialRing(K)
sage: H = HyperellipticCurve(t^5 + 27*t + 3, t)
sage: H.frobenius_polynomial() # long time, 3s on a Corei7
x^4 + 2*x^3 - 58*x^2 + 202*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


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


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.

p_rank()

INPUT:

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

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 #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()

Gives 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)


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)
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