Operations on Curves#

Quadratic Twists#

There are standard functions for quadratic twists of hyperelliptic curves in characteristic not equal to 2. In addition, from the new package [Lercier et al., 2021] (described in more detail in the next section) there are functions to return all twists of a genus 2 hyperelliptic curve over a finite field of any characteristic.

QuadraticTwist(C, d): CrvHyp, RngElt -> CrvHyp#: Given a hyperelliptic curve \(C\) defined over a field \(k\) of characteristic not equal to 2 and an element \(d\) that is coercible into \(k\), return the quadratic twist of \(C\) by \(d\).

QuadraticTwist(C): CrvHyp -> CrvHyp#: Given a hyperelliptic curve \(C\) defined over a finite field \(k\), return the standard quadratic twist of \(C\) over the unique extension of \(k\) of degree 2. If the characteristic of \(k\) is odd, then this is the same as the twist of \(C\) by a primitive element of \(k\).

QuadraticTwists(C): CrvHyp -> SeqEnum#: Given a hyperelliptic curve \(C\) defined over a finite field \(k\) of odd characteristic, return a sequence containing the non-isomorphic quadratic twists of \(C\).

IsQuadraticTwist(C, D): CrvHyp, CrvHyp -> BoolElt, RngElt#

verbose: CrvHypIso                    Default: Verbose : 3

Given hyperelliptic curves \(C\) and \(D\) over a common field \(k\) having characteristic not equal to two, return true if and only if \(C\) is a quadratic twist of \(D\) over \(k\). If so, the twisting factor is returned as the second value.

Example: Quadratic Twists (ex-a5b80e)#

We construct the quadratic twists of the hyperelliptic curve \(y^2=x^6+x^2+1\) defined over \({\bf F}_{7}\).

> P<x> := PolynomialRing(GF(7));
> C := HyperellipticCurve(x^6+x^2+1);
> QuadraticTwists(C);
[
    Hyperelliptic Curve defined by y^2 = x^6 + x^2 + 1 over GF(7),
    Hyperelliptic Curve defined by y^2 = 3*x^6 + 3*x^2 + 3 over GF(7)
]
> IsIsomorphic($1[1],$1[2]);
false

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Example: Quadratic Twists (ex-0bc2c3)#

We take a hyperelliptic curve over the rationals and form a quadratic twist of it.

> P<x> := PolynomialRing(Rationals());
> C := HyperellipticCurve(x^6+x);
> C7 := QuadraticTwist(C, 7);
> C7;
Hyperelliptic Curve defined by y^2 = 7*x^6 + 7*x over Rational Field

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We now use the function IsIsomorphic to verify that \(C\) and \(C_7\) are nonisomorphic. We then extend the field of definition of both curves to \({\mathbb{Q}}(\sqrt{7})\) and verify that the curves become isomorphic over this extension.

> IsIsomorphic(C, C7);
false
> K<w> := ext< Rationals() | x^2-7 >;
> CK := BaseChange(C, K);
> C7K := BaseChange(C7, K);
> IsIsomorphic(CK, C7K);
true (x : y : z) :-> (x : -1/7*w*y : z)

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Elementary Invariants#

HyperellipticPolynomials(C): CrvHyp -> RngUPolElt, RngUPolElt#: The univariate polynomials \(f(x)\), \(h(x)\), in that order, defining the hyperelliptic curve \(C\) by \(y^2 + h(x)y = f(x)\).

Degree(C): CrvHyp -> RngIntElt#
Degree(C): SetPtHyp -> RngIntElt#: The degree of the hyperelliptic curve \(C\) or a pointset \(C\) of a hyperelliptic curve.

Discriminant(C): CrvHyp -> RngElt#: The discriminant of the hyperelliptic curve \(C\).

Genus(C): CrvHyp -> RngIntElt#: The genus of the hyperelliptic curve \(C\).

Conductor(C): CrvHyp -> RngIntElt#: Conductor of a hyperelliptic curve \(C\) defined over \({\mathbb{Q}}\) or a number field.

Conductor(C, p): CrvHyp[FldRat], RngIntElt -> RngIntElt#
Conductor(C, P): CrvHyp[FldNum], RngOrdIdl -> RngIntElt#: Conductor exponent of a hyperelliptic curve \(C/{\mathbb{Q}}\) at a prime \(p\) or a curve over a number field at a prime ideal \(P\).

Example: Crvhyp Conductor Q (ex-9451e3)#

> R<x>:=PolynomialRing(Rationals());
> C:=HyperellipticCurve(x^8+1,x); C;
Hyperelliptic Curve defined by y^2 + x*y = x^8 + 1 over Rational Field
> Factorization(Integers()!Discriminant(C));
[ <2, 4>, <109, 2>, <601, 2> ]
> Factorization(Conductor(C));                       // global
[ <2, 2>, <109, 2>, <601, 2> ]
> Conductor(C,2),Conductor(C,3),Conductor(C,5);      // local
2 0 0

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ConductorExponent(C): CrvHyp[FldPad] -> RngIntElt#
Conductor(C): CrvHyp[FldPad] -> FldPadElt#: Conductor exponent and conductor ideal (uniformizer to the conductor exponent power) of a hyperelliptic curve over a \(p\)-adic field.

Example: Crvhyp Conductor Padic (ex-581c7e)#

Here is a genus 4 example over \({\mathbb{Q}}_3\).

> K:=pAdicField(3,20);
> R<x>:=PolynomialRing(K);
> C:=HyperellipticCurve(x^9+1);
> ConductorExponent(C);
12
> Conductor(C);
3^12 + O(3^32)

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EulerFactor(C, p): CrvHyp[FldRat], RngIntElt -> RngUPolElt#

EulerFactor(C, P): CrvHyp[FldNum], RngOrdIdl -> RngUPolElt#

EulerFactor(C): CrvHyp[FldPad] -> RngUPolElt#

Degree: RngIntElt                    Default: Infinity()

The Euler factor (local polynomial) of a hyperelliptic curve defined over \({\mathbb{Q}}\), a number field or a \(p\)-adic field at a prime \(p\). If Degree is specified, it is computed only up to that degree.

EulerFactor(C, p): CrvHyp[FldNum], RngIntElt -> RngUPolElt#

Degree: RngIntElt                    Default: Infinity()

The total Euler factor of a hyperelliptic curve defined over a number field over all primes above \(p\). If Degree is specified, it is computed only up to that degree.

Example: Crvhyp Eulerfactor (ex-9f8b78)#

We consider the genus 2 curve \(C/{\mathbb{Q}}: y^2=x^5+x^2+3\) and compute its conductor at \(p=3\), and over \(K={\mathbb{Q}}(\root 5\of 3)\) at the unique prime ideal \(P|3\).

> R<x>:=PolynomialRing(Rationals());
> C:=HyperellipticCurve(x^5+x^2+3);
> EulerFactor(C,3);                     // local factor /Q at 3
-T + 1
> K:=NumberField(x^5-3);
> P:=Ideal(Decomposition(K,5)[1,1]);
> EulerFactor(BaseChange(C,K),P);       // local factor /K at P
25*x^4 - 5*x^3 - x + 1
> EulerFactor(BaseChange(C,Completion(K,P)));   // same, computed over K_P
25*x^4 - 5*x^3 - x + 1
> EulerFactor(BaseChange(C,K),2);       // total factor /K over all P|2
1

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Base Ring#

BaseField(C): Sch -> Fld#
BaseRing(C): Sch -> Fld#
CoefficientRing(C): Sch -> Fld#: The base field of the hyperelliptic curve \(C\).