Hyperelliptic curves over a finite field - Elliptic curves

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# 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)]

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: `HyperellipticCurve_generic`, `ProjectivePlaneCurve_finite_field`

  Cartier_matrix()

    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:

    14. 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]

  Hasse_Witt()

    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]

  a_number()

    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

  cardinality(extension_degree=1)

    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

    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

  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

        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]

  cardinality_hypellfrob(extension_degree=1, algorithm=None)

    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

  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

        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]

    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]

  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

        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]

    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]

    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]

  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

        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))

    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]

    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]

  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

        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]

    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

    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

  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

        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]

  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

        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)]

    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.

    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

  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

        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)]

    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.

    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

  frobenius_polynomial(algorithm=None, **kwargs)

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

    INPUT:

    - `algorithm` – the algorithm to use, one of

      - `None` (default) – automatically select an algorithm.

      - `'pari'` – use the PARI function `hyperellcharpoly`.

      - `'cardinalities'` – compute 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.

      - `'matrix'` – compute the charpoly of the frobenius matrix.

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

    - `**kwargs` – additional keyword arguments passed to the internal method. Undocumented feature, may be removed in the future.

    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

    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

    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

    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

    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

    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

    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

  frobenius_polynomial_cardinalities(*args, **kwds)

    Deprecated: Use `frobenius_polynomial(algorithm="cardinalities")` instead. See Issue #40528 for details.

  frobenius_polynomial_matrix(*args, **kwds)

    Deprecated: Use `frobenius_polynomial(algorithm="matrix")` instead. See Issue #40528 for details.

  frobenius_polynomial_pari(*args, **kwds)

    Deprecated: Use `frobenius_polynomial(algorithm="pari")` instead. See Issue #40528 for details.

  p_rank()

    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

  points()

    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

        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

    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)]

    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)]

    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

        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)

    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)