# Properties

 Label 129600.b.129600.1 Conductor $129600$ Discriminant $129600$ Mordell-Weil group trivial Sato-Tate group $\mathrm{USp}(4)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q$$ $$\End(J) \otimes \Q$$ $$\Q$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type no

# Related objects

Show commands: Magma / SageMath

## Simplified equation

 $y^2 + xy = -x^6 - 2x^5 - 2x^4 - x^3 + x^2 + 2x - 1$ (homogenize, simplify) $y^2 + xz^2y = -x^6 - 2x^5z - 2x^4z^2 - x^3z^3 + x^2z^4 + 2xz^5 - z^6$ (dehomogenize, simplify) $y^2 = -4x^6 - 8x^5 - 8x^4 - 4x^3 + 5x^2 + 8x - 4$ (homogenize, minimize)

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

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

sage: X = HyperellipticCurve(R([-4, 8, 5, -4, -8, -8, -4]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$129600$$ $$=$$ $$2^{6} \cdot 3^{4} \cdot 5^{2}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$129600$$ $$=$$ $$2^{6} \cdot 3^{4} \cdot 5^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$708$$ $$=$$ $$2^{2} \cdot 3 \cdot 59$$ $$I_4$$ $$=$$ $$16335$$ $$=$$ $$3^{3} \cdot 5 \cdot 11^{2}$$ $$I_6$$ $$=$$ $$3582090$$ $$=$$ $$2 \cdot 3^{3} \cdot 5 \cdot 13267$$ $$I_{10}$$ $$=$$ $$-16200$$ $$=$$ $$- 2^{3} \cdot 3^{4} \cdot 5^{2}$$ $$J_2$$ $$=$$ $$708$$ $$=$$ $$2^{2} \cdot 3 \cdot 59$$ $$J_4$$ $$=$$ $$9996$$ $$=$$ $$2^{2} \cdot 3 \cdot 7^{2} \cdot 17$$ $$J_6$$ $$=$$ $$-220864$$ $$=$$ $$- 2^{6} \cdot 7 \cdot 17 \cdot 29$$ $$J_8$$ $$=$$ $$-64072932$$ $$=$$ $$- 2^{2} \cdot 3 \cdot 7 \cdot 11 \cdot 17 \cdot 4079$$ $$J_{10}$$ $$=$$ $$-129600$$ $$=$$ $$- 2^{6} \cdot 3^{4} \cdot 5^{2}$$ $$g_1$$ $$=$$ $$-34316366352/25$$ $$g_2$$ $$=$$ $$-684322828/25$$ $$g_3$$ $$=$$ $$192206896/225$$

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

This curve has no rational points.
This curve has no rational points.
This curve has no rational points.

magma: []; // minimal model

magma: []; // simplified model

Number of rational Weierstrass points: $$0$$

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

This curve is locally solvable except over $\Q_{3}$.

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: trivial

magma: MordellWeilGroupGenus2(Jacobian(C));

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$0$$ Regulator: $$1$$ Real period: $$2.045372$$ Tamagawa product: $$1$$ Torsion order: $$1$$ Leading coefficient: $$2.045372$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$6$$ $$6$$ $$1$$ $$1 + T$$
$$3$$ $$4$$ $$4$$ $$1$$ $$1 - 3 T + 3 T^{2}$$
$$5$$ $$2$$ $$2$$ $$1$$ $$1 + 2 T + 5 T^{2}$$

## Galois representations

The mod-$\ell$ Galois representation has maximal image $$\GSp(4,\F_\ell)$$ for all primes $$\ell$$ except those listed.

Prime $$\ell$$ mod-$$\ell$$ image Is torsion prime?
$$2$$ 2.40.1 no
$$3$$ 3.40.1 no

## Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $\mathrm{USp}(4)$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{USp}(4)$$

## Decomposition of the Jacobian

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

magma: HeuristicDecompositionFactors(C);

## Endomorphisms of the Jacobian

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

Endomorphism ring over $$\Q$$:

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

All $$\overline{\Q}$$-endomorphisms of the Jacobian are defined over $$\Q$$.

magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

magma: HeuristicIsGL2(C); HeuristicEndomorphismDescription(C); HeuristicEndomorphismFieldOfDefinition(C);

magma: HeuristicIsGL2(C : Geometric := true); HeuristicEndomorphismDescription(C : Geometric := true); HeuristicEndomorphismLatticeDescription(C);