# Properties

 Label 708.a.181248.1 Conductor $708$ Discriminant $-181248$ Mordell-Weil group $$\Z/{2}\Z$$ 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

# Learn more

Show commands: Magma / SageMath

## Simplified equation

 $y^2 + (x^3 + 1)y = -x^6 - 4x^5 + 9x^4 + 48x^3 - 41x^2 - 98x - 36$ (homogenize, simplify) $y^2 + (x^3 + z^3)y = -x^6 - 4x^5z + 9x^4z^2 + 48x^3z^3 - 41x^2z^4 - 98xz^5 - 36z^6$ (dehomogenize, simplify) $y^2 = -3x^6 - 16x^5 + 36x^4 + 194x^3 - 164x^2 - 392x - 143$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-36, -98, -41, 48, 9, -4, -1]), R([1, 0, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-36, -98, -41, 48, 9, -4, -1], R![1, 0, 0, 1]);

sage: X = HyperellipticCurve(R([-143, -392, -164, 194, 36, -16, -3]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$708$$ $$=$$ $$2^{2} \cdot 3 \cdot 59$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-181248$$ $$=$$ $$- 2^{10} \cdot 3 \cdot 59$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$234100$$ $$=$$ $$2^{2} \cdot 5^{2} \cdot 2341$$ $$I_4$$ $$=$$ $$3468879025$$ $$=$$ $$5^{2} \cdot 138755161$$ $$I_6$$ $$=$$ $$202585466081177$$ $$=$$ $$1069463 \cdot 189427279$$ $$I_{10}$$ $$=$$ $$-23199744$$ $$=$$ $$- 2^{17} \cdot 3 \cdot 59$$ $$J_2$$ $$=$$ $$58525$$ $$=$$ $$5^{2} \cdot 2341$$ $$J_4$$ $$=$$ $$-1820975$$ $$=$$ $$- 5^{2} \cdot 13^{2} \cdot 431$$ $$J_6$$ $$=$$ $$60952909$$ $$=$$ $$109 \cdot 559201$$ $$J_8$$ $$=$$ $$62829762150$$ $$=$$ $$2 \cdot 3 \cdot 5^{2} \cdot 47 \cdot 8912023$$ $$J_{10}$$ $$=$$ $$-181248$$ $$=$$ $$- 2^{10} \cdot 3 \cdot 59$$ $$g_1$$ $$=$$ $$-686605237334059580078125/181248$$ $$g_2$$ $$=$$ $$365029741228054296875/181248$$ $$g_3$$ $$=$$ $$-208774418179643125/181248$$

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 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: $$\Z/{2}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - D_\infty$$ $$x^2 + 2xz - 11z^2$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$-15xz^2 + 21z^3$$ $$0$$ $$2$$
Generator $D_0$ Height Order
$$D_0 - D_\infty$$ $$x^2 + 2xz - 11z^2$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$-15xz^2 + 21z^3$$ $$0$$ $$2$$
Generator $D_0$ Height Order
$$D_0 - D_\infty$$ $$x^2 + 2xz - 11z^2$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$x^3 - 30xz^2 + 43z^3$$ $$0$$ $$2$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$3$$ Regulator: $$1$$ Real period: $$0.325343$$ Tamagawa product: $$1$$ Torsion order: $$2$$ Leading coefficient: $$0.325343$$ Analytic order of Ш: $$4$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$2$$ $$10$$ $$1$$ $$1 + T^{2}$$
$$3$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + T + 3 T^{2} )$$
$$59$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 59 T^{2} )$$

## 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?
$$2$$ 2.15.1 yes
$$5$$ not computed 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);