# 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](../../references.md#cite-lrs)] (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` ```magma 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 ${\bf F}_{7}$. ```magma > P := 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 ``` ### `Example: Quadratic Twists (ex-0bc2c3)` We take a hyperelliptic curve over the rationals and form a quadratic twist of it. ```magma > P := 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 $C_7$ 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. ```magma > IsIsomorphic(C, C7); false > K := 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) ``` ## 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)` ```magma > R:=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 ``` ### `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 ${\mathbb{Q}}_3$. ```magma > K:=pAdicField(3,20); > R:=PolynomialRing(K); > C:=HyperellipticCurve(x^9+1); > ConductorExponent(C); 12 > Conductor(C); 3^12 + O(3^32) ``` ### `EulerFactor(C, p): CrvHyp[FldRat], RngIntElt -> RngUPolElt` ### `EulerFactor(C, P): CrvHyp[FldNum], RngOrdIdl -> RngUPolElt` ### `EulerFactor(C): CrvHyp[FldPad] -> RngUPolElt` ```magma 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` ```magma 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 $P|3$. ```magma > R:=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 ``` ## Base Ring ### `BaseField(C): Sch -> Fld` ### `BaseRing(C): Sch -> Fld` ### `CoefficientRing(C): Sch -> Fld` The base field of the hyperelliptic curve $C$.