# Properties

 Label 12500.a.12500.1 Conductor 12500 Discriminant 12500 Mordell-Weil group $$\Z/{5}\Z$$ Sato-Tate group $N(G_{3,3})$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathrm{RM}$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type no

# Related objects

Show commands for: SageMath / Magma

## Simplified equation

 $y^2 + (x^3 + 1)y = x^6 + 2x^3 - x$ (homogenize, simplify) $y^2 + (x^3 + z^3)y = x^6 + 2x^3z^3 - xz^5$ (dehomogenize, simplify) $y^2 = 5x^6 + 10x^3 - 4x + 1$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, 0, 2, 0, 0, 1]), R([1, 0, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, 0, 2, 0, 0, 1], R![1, 0, 0, 1]);

sage: X = HyperellipticCurve(R([1, -4, 0, 10, 0, 0, 5]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$12500$$ $$=$$ $$2^{2} \cdot 5^{5}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$12500$$ $$=$$ $$2^{2} \cdot 5^{5}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$60$$ $$=$$ $$2^{2} \cdot 3 \cdot 5$$ $$I_4$$ $$=$$ $$2025$$ $$=$$ $$3^{4} \cdot 5^{2}$$ $$I_6$$ $$=$$ $$67995$$ $$=$$ $$3^{2} \cdot 5 \cdot 1511$$ $$I_{10}$$ $$=$$ $$-512$$ $$=$$ $$- 2^{9}$$ $$J_2$$ $$=$$ $$75$$ $$=$$ $$3 \cdot 5^{2}$$ $$J_4$$ $$=$$ $$-1875$$ $$=$$ $$- 3 \cdot 5^{4}$$ $$J_6$$ $$=$$ $$-73125$$ $$=$$ $$- 3^{2} \cdot 5^{4} \cdot 13$$ $$J_8$$ $$=$$ $$-2250000$$ $$=$$ $$- 2^{4} \cdot 3^{2} \cdot 5^{6}$$ $$J_{10}$$ $$=$$ $$-12500$$ $$=$$ $$- 2^{2} \cdot 5^{5}$$ $$g_1$$ $$=$$ $$-759375/4$$ $$g_2$$ $$=$$ $$253125/4$$ $$g_3$$ $$=$$ $$131625/4$$

sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]

magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);

## Automorphism group

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

## Rational points

All points: $$(0 : 0 : 1),\, (-1 : 0 : 1),\, (0 : -1 : 1)$$

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

Number of rational Weierstrass points: $$1$$

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

This curve is locally solvable everywhere.

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);

## Mordell-Weil group of the Jacobian

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

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(-1 : 0 : 1) + (0 : 0 : 1) - D_\infty$$ $$x (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$5$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$0$$ Regulator: $$1$$ Real period: $$25.01138$$ Tamagawa product: $$1$$ Torsion order: $$5$$ Leading coefficient: $$1.000455$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$2$$ $$2$$ $$1$$ $$1 + T^{2}$$
$$5$$ $$5$$ $$5$$ $$1$$ $$1$$

## Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $N(G_{3,3})$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{SU}(2)\times\mathrm{SU}(2)$$

## 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 $$K = \Q (a) \simeq$$ $$\Q(\sqrt{5})$$ with defining polynomial $$x^{2} - x - 1$$

Of $$\GL_2$$-type over $$\overline{\Q}$$

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

 $$\End (J_{\overline{\Q}})$$ $$\simeq$$ $$\Z [\frac{1 + \sqrt{5}}{2}]$$ $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\simeq$$ $$\Q(\sqrt{5})$$ $$\End (J_{\overline{\Q}}) \otimes \R$$ $$\simeq$$ $$\R \times \R$$