Show commands for: Magma / SageMath

## Simplified equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, 0, 0, 0, 1], R!);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, 0, 0, 0, 1]), R())

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, 0, 0, 0, 1], R!);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, 0, 0, 0, 0, 4]))

 $y^2 + y = x^5$ (homogenize, simplify) $y^2 + z^3y = x^5z$ (dehomogenize, simplify) $y^2 = 4x^5 + 1$ (minimize, homogenize)

## Invariants

 $$N$$ = $$3125$$ = $$5^{5}$$ magma: Conductor(LSeries(C)); Factorization($1); $$\Delta$$ = $$3125$$ = $$5^{5}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$0$$ = $$0$$ $$I_4$$ = $$0$$ = $$0$$ $$I_6$$ = $$0$$ = $$0$$ $$I_{10}$$ = $$12800000$$ = $$2^{12} \cdot 5^{5}$$ $$J_2$$ = $$0$$ = $$0$$ $$J_4$$ = $$0$$ = $$0$$ $$J_6$$ = $$0$$ = $$0$$ $$J_8$$ = $$0$$ = $$0$$ $$J_{10}$$ = $$3125$$ = $$5^{5}$$ $$g_1$$ = $$0$$ $$g_2$$ = $$0$$ $$g_3$$ = $$0$$

## Automorphism group

 magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X)$$ $$\simeq$$$C_2$magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$ $C_{10}$

## Rational points

magma: [C![0,-1,1],C![0,0,1],C![1,0,0]];

Points: $$(0 : 0 : 1),\, (1 : 0 : 0),\, (0 : -1 : 1)$$

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));

Number of rational Weierstrass points: $$1$$

magma: f,h:=HyperellipticPolynomials(C); g:=4*f+h^2; HasPointsEverywhereLocally(g,2) and (#Roots(ChangeRing(g,RealField())) gt 0 or LeadingCoefficient(g) gt 0);

This curve is locally solvable everywhere.

## Mordell-Weil group of the Jacobian:

magma: MordellWeilGroupGenus2(Jacobian(C));

Group structure: $$\Z/{5}\Z$$

Generator Height Order
$$x^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$5$$

## BSD invariants

 Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$0$$ Regulator: $$1$$ Real period: $$17.95784$$ Tamagawa product: $$1$$ Torsion order: $$5$$ Leading coefficient: $$0.718313$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$5$$ $$5$$ $$5$$ $$1$$ $$1$$

## Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $F_{ac}$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{U}(1)\times\mathrm{U}(1)$$

## Decomposition of the Jacobian

Simple over $$\overline{\Q}$$

## Endomorphisms of the Jacobian

Not of $$\GL_2$$-type over $$\Q$$

Endomorphism ring over $$\Q$$:

 $$\End (J_{})$$ $$\simeq$$ $$\Z$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R$$

Smallest field over which all endomorphisms are defined:
Galois number field $$K = \Q (a) \simeq$$ $$\Q(\zeta_{5})$$ with defining polynomial $$x^{4} - x^{3} + x^{2} - x + 1$$

Not of $$\GL_2$$-type over $$\overline{\Q}$$

Endomorphism ring over $$\overline{\Q}$$:

 $$\End (J_{\overline{\Q}})$$ $$\simeq$$ the maximal order of $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\simeq$$ $$\Q(\zeta_{5})$$ (CM) $$\End (J_{\overline{\Q}}) \otimes \R$$ $$\simeq$$ $$\C \times \C$$

### Remainder of the endomorphism lattice by field

Over subfield $$F \simeq$$ $$\Q(\sqrt{5})$$ with generator $$a^{3} - a^{2}$$ with minimal polynomial $$x^{2} - x - 1$$:

 $$\End (J_{F})$$ $$\simeq$$ $$\Z [\frac{1 + \sqrt{5}}{2}]$$ $$\End (J_{F}) \otimes \Q$$ $$\simeq$$ $$\Q(\sqrt{5})$$ $$\End (J_{F}) \otimes \R$$ $$\simeq$$ $$\R \times \R$$
Sato Tate group: $F_{ab}$
Of $$\GL_2$$-type, simple