# Properties

 Label 1408.b.720896.2 Conductor 1408 Discriminant -720896 Mordell-Weil group $$\Z/{2}\Z \times \Z/{4}\Z$$ Sato-Tate group $\mathrm{USp}(4)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type no

# Related objects

Show commands for: SageMath / Magma

## Simplified equation

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

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, -4, 2, 0, 1]), R([]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, -4, 2, 0, 1], R![]);

sage: X = HyperellipticCurve(R([0, 1, -4, 2, 0, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$1408$$ $$=$$ $$2^{7} \cdot 11$$ magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(1408,2),R![1]>*])); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-720896$$ $$=$$ $$- 2^{16} \cdot 11$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$1024$$ $$=$$ $$2^{10}$$ $$I_4$$ $$=$$ $$-81920$$ $$=$$ $$- 2^{14} \cdot 5$$ $$I_6$$ $$=$$ $$-40632320$$ $$=$$ $$- 2^{18} \cdot 5 \cdot 31$$ $$I_{10}$$ $$=$$ $$-2952790016$$ $$=$$ $$- 2^{28} \cdot 11$$ $$J_2$$ $$=$$ $$128$$ $$=$$ $$2^{7}$$ $$J_4$$ $$=$$ $$1536$$ $$=$$ $$2^{9} \cdot 3$$ $$J_6$$ $$=$$ $$45056$$ $$=$$ $$2^{12} \cdot 11$$ $$J_8$$ $$=$$ $$851968$$ $$=$$ $$2^{16} \cdot 13$$ $$J_{10}$$ $$=$$ $$-720896$$ $$=$$ $$- 2^{16} \cdot 11$$ $$g_1$$ $$=$$ $$-524288/11$$ $$g_2$$ $$=$$ $$-49152/11$$ $$g_3$$ $$=$$ $$-1024$$

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: $$(1 : 0 : 0),\, (0 : 0 : 1),\, (1 : 0 : 1)$$

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

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
$$(0 : 0 : 1) + (1 : 0 : 1) - 2 \cdot(1 : 0 : 0)$$ $$(-x + z) x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$2$$
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$x^2 - 4xz + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$8xz^2 - 2z^3$$ $$0$$ $$4$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$2$$ Regulator: $$1$$ Real period: $$7.656364$$ Tamagawa product: $$4$$ Torsion order: $$8$$ Leading coefficient: $$0.478522$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$7$$ $$16$$ $$4$$ $$1$$
$$11$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 4 T + 11 T^{2} )$$

## 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 $$\Q$$.