# Properties

 Label 12544.b.200704.1 Conductor $12544$ Discriminant $200704$ Mordell-Weil group $$\Z/{2}\Z \times \Z/{4}\Z$$ 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: SageMath / Magma

## Simplified equation

 $y^2 = x^5 - 7x^4 + 8x^3 + 15x^2 + 4x$ (homogenize, simplify) $y^2 = x^5z - 7x^4z^2 + 8x^3z^3 + 15x^2z^4 + 4xz^5$ (dehomogenize, simplify) $y^2 = x^5 - 7x^4 + 8x^3 + 15x^2 + 4x$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 4, 15, 8, -7, 1]), R([]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 4, 15, 8, -7, 1], R![]);

sage: X = HyperellipticCurve(R([0, 4, 15, 8, -7, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$12544$$ $$=$$ $$2^{8} \cdot 7^{2}$$ magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(12544,2),R![1]>*])); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$200704$$ $$=$$ $$2^{12} \cdot 7^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$2224$$ $$=$$ $$2^{4} \cdot 139$$ $$I_4$$ $$=$$ $$9940$$ $$=$$ $$2^{2} \cdot 5 \cdot 7 \cdot 71$$ $$I_6$$ $$=$$ $$7386820$$ $$=$$ $$2^{2} \cdot 5 \cdot 7 \cdot 19 \cdot 2777$$ $$I_{10}$$ $$=$$ $$784$$ $$=$$ $$2^{4} \cdot 7^{2}$$ $$J_2$$ $$=$$ $$4448$$ $$=$$ $$2^{5} \cdot 139$$ $$J_4$$ $$=$$ $$797856$$ $$=$$ $$2^{5} \cdot 3 \cdot 8311$$ $$J_6$$ $$=$$ $$183931136$$ $$=$$ $$2^{8} \cdot 743 \cdot 967$$ $$J_8$$ $$=$$ $$45387874048$$ $$=$$ $$2^{8} \cdot 11 \cdot 17 \cdot 83 \cdot 11423$$ $$J_{10}$$ $$=$$ $$200704$$ $$=$$ $$2^{12} \cdot 7^{2}$$ $$g_1$$ $$=$$ $$425073415774208/49$$ $$g_2$$ $$=$$ $$17141897862912/49$$ $$g_3$$ $$=$$ $$888433369664/49$$

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: $$(1 : 0 : 0),\, (0 : 0 : 1),\, (4 : 0 : 1)$$
All points: $$(1 : 0 : 0),\, (0 : 0 : 1),\, (4 : 0 : 1)$$
All points: $$(1 : 0 : 0),\, (0 : 0 : 1),\, (4 : 0 : 1)$$

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

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

Number of rational Weierstrass points: $$3$$

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: $$\Z/{2}\Z \times \Z/{4}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(4 : 0 : 1) - (1 : 0 : 0)$$ $$x - 4z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$2$$
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$x^2 - 4xz - 2z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-xz^2$$ $$0$$ $$4$$
Generator $D_0$ Height Order
$$(4 : 0 : 1) - (1 : 0 : 0)$$ $$x - 4z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$2$$
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$x^2 - 4xz - 2z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-xz^2$$ $$0$$ $$4$$
Generator $D_0$ Height Order
$$(4 : 0 : 1) - (1 : 0 : 0)$$ $$x - 4z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$2$$
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$x^2 - 4xz - 2z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-1/2xz^2$$ $$0$$ $$4$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$2$$ Regulator: $$1$$ Real period: $$13.95491$$ Tamagawa product: $$4$$ Torsion order: $$8$$ Leading coefficient: $$0.872182$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$8$$ $$12$$ $$4$$ $$1$$
$$7$$ $$2$$ $$2$$ $$1$$ $$1 + 7 T^{2}$$

## Galois representations

For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not the maximal image $\GSp(4,\F_\ell)$

Prime $$\ell$$ mod-$$\ell$$ image
$$2$$ 2.240.1

## Sato-Tate group

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

## Decomposition of the Jacobian

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

## Endomorphisms of the Jacobian

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

Endomorphism ring over $$\Q$$:

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

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