# Properties

 Label 32400.e.486000.1 Conductor $32400$ Discriminant $-486000$ 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

# Related objects

Show commands: Magma / SageMath

## Simplified equation

 $y^2 + x^2y = x^5 - 6x^4 + 7x^3 + 12x - 21$ (homogenize, simplify) $y^2 + x^2zy = x^5z - 6x^4z^2 + 7x^3z^3 + 12xz^5 - 21z^6$ (dehomogenize, simplify) $y^2 = 4x^5 - 23x^4 + 28x^3 + 48x - 84$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-21, 12, 0, 7, -6, 1]), R([0, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-21, 12, 0, 7, -6, 1], R![0, 0, 1]);

sage: X = HyperellipticCurve(R([-84, 48, 0, 28, -23, 4]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$32400$$ $$=$$ $$2^{4} \cdot 3^{4} \cdot 5^{2}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-486000$$ $$=$$ $$- 2^{4} \cdot 3^{5} \cdot 5^{3}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$3096$$ $$=$$ $$2^{3} \cdot 3^{2} \cdot 43$$ $$I_4$$ $$=$$ $$252900$$ $$=$$ $$2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 281$$ $$I_6$$ $$=$$ $$172934100$$ $$=$$ $$2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 192149$$ $$I_{10}$$ $$=$$ $$-1944000$$ $$=$$ $$- 2^{6} \cdot 3^{5} \cdot 5^{3}$$ $$J_2$$ $$=$$ $$1548$$ $$=$$ $$2^{2} \cdot 3^{2} \cdot 43$$ $$J_4$$ $$=$$ $$57696$$ $$=$$ $$2^{5} \cdot 3 \cdot 601$$ $$J_6$$ $$=$$ $$7496356$$ $$=$$ $$2^{2} \cdot 7 \cdot 267727$$ $$J_8$$ $$=$$ $$2068882668$$ $$=$$ $$2^{2} \cdot 3^{3} \cdot 19156321$$ $$J_{10}$$ $$=$$ $$-486000$$ $$=$$ $$- 2^{4} \cdot 3^{5} \cdot 5^{3}$$ $$g_1$$ $$=$$ $$-2286275305536/125$$ $$g_2$$ $$=$$ $$-55046830464/125$$ $$g_3$$ $$=$$ $$-13860762244/375$$

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: $$(1 : 0 : 0),\, (7 : -98 : 4)$$
All points: $$(1 : 0 : 0),\, (7 : -98 : 4)$$
All points: $$(1 : 0 : 0),\, (7 : 0 : 4)$$

magma: [C![1,0,0],C![7,-98,4]]; // minimal model

magma: [C![1,0,0],C![7,0,4]]; // simplified model

Number of rational Weierstrass points: $$2$$

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
$$(7 : -98 : 4) - (1 : 0 : 0)$$ $$4x - 7z$$ $$=$$ $$0,$$ $$32y$$ $$=$$ $$-49z^3$$ $$0$$ $$2$$
Generator $D_0$ Height Order
$$(7 : -98 : 4) - (1 : 0 : 0)$$ $$4x - 7z$$ $$=$$ $$0,$$ $$32y$$ $$=$$ $$-49z^3$$ $$0$$ $$2$$
Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$4x - 7z$$ $$=$$ $$0,$$ $$32y$$ $$=$$ $$x^2z - 98z^3$$ $$0$$ $$2$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$1$$ Regulator: $$1$$ Real period: $$2.342976$$ Tamagawa product: $$4$$ Torsion order: $$2$$ Leading coefficient: $$2.342976$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

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

## 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?
$$2$$ 2.30.3 yes
$$3$$ 3.2160.24 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);