There are standard functions for quadratic twists of hyperelliptic curves in characteristic not equal to 2. In addition, from the new package [LRS21] (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.
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.
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.
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.
SetVerbose("CrvHypIso", n): Maximum: 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.
> 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
> 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
We now use the function IsIsomorphic to verify that C and C7 are nonisomorphic. We then extend the field of definition of both curves to 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)
The univariate polynomials f(x), h(x), in that order, defining the hyperelliptic curve C by y2 + h(x)y = f(x).
The degree of the hyperelliptic curve C or a pointset C of a hyperelliptic curve.
The discriminant of the hyperelliptic curve C.
The genus of the hyperelliptic curve C.
Conductor of a hyperelliptic curve C defined over Q or a number field.
Conductor exponent of a hyperelliptic curve C/Q at a prime p or a curve over a number field at a prime ideal P.
> 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
Conductor exponent and conductor ideal (uniformizer to the conductor exponent power) of a hyperelliptic curve over a p-adic field.
> K:=pAdicField(3,20); > R<x>:=PolynomialRing(K); > C:=HyperellipticCurve(x^9+1); > ConductorExponent(C); 12 > Conductor(C); 3^12 + O(3^32)
Degree: RngIntElt Default: ∞
The Euler factor (local polynomial) of a hyperelliptic curve defined over 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.
Degree: RngIntElt Default: ∞
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.
> 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
The base field of the hyperelliptic curve C.