# Properties

 Label 7549.a.7549.1 Conductor 7549 Discriminant -7549 Mordell-Weil group $$\Z \times \Z$$ 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

 $y^2 + (x^3 + x + 1)y = -2x^4 + 3x^2 - x - 2$ (homogenize, simplify) $y^2 + (x^3 + xz^2 + z^3)y = -2x^4z^2 + 3x^2z^4 - xz^5 - 2z^6$ (dehomogenize, simplify) $y^2 = x^6 - 6x^4 + 2x^3 + 13x^2 - 2x - 7$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-2, -1, 3, 0, -2]), R([1, 1, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-2, -1, 3, 0, -2], R![1, 1, 0, 1]);

sage: X = HyperellipticCurve(R([-7, -2, 13, 2, -6, 0, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$7549$$ $$=$$ $$7549$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-7549$$ $$=$$ $$-7549$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$2952$$ $$=$$ $$2^{3} \cdot 3^{2} \cdot 41$$ $$I_4$$ $$=$$ $$1380$$ $$=$$ $$2^{2} \cdot 3 \cdot 5 \cdot 23$$ $$I_6$$ $$=$$ $$279144$$ $$=$$ $$2^{3} \cdot 3^{2} \cdot 3877$$ $$I_{10}$$ $$=$$ $$-30920704$$ $$=$$ $$- 2^{12} \cdot 7549$$ $$J_2$$ $$=$$ $$369$$ $$=$$ $$3^{2} \cdot 41$$ $$J_4$$ $$=$$ $$5659$$ $$=$$ $$5659$$ $$J_6$$ $$=$$ $$117293$$ $$=$$ $$11 \cdot 10663$$ $$J_8$$ $$=$$ $$2814209$$ $$=$$ $$953 \cdot 2953$$ $$J_{10}$$ $$=$$ $$-7549$$ $$=$$ $$-7549$$ $$g_1$$ $$=$$ $$-6841192812849/7549$$ $$g_2$$ $$=$$ $$-284327451531/7549$$ $$g_3$$ $$=$$ $$-15970732173/7549$$

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

Known points
$$(1 : 0 : 0)$$ $$(1 : -1 : 0)$$ $$(-1 : 0 : 1)$$ $$(-1 : 1 : 1)$$ $$(1 : -1 : 1)$$ $$(1 : -2 : 1)$$
$$(2 : -3 : 1)$$ $$(-2 : 4 : 1)$$ $$(-2 : 5 : 1)$$ $$(2 : -8 : 1)$$

magma: [C![-2,4,1],C![-2,5,1],C![-1,0,1],C![-1,1,1],C![1,-2,1],C![1,-1,0],C![1,-1,1],C![1,0,0],C![2,-8,1],C![2,-3,1]];

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: $$\Z \times \Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2 - 2z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-2xz^2 - z^3$$ $$0.152213$$ $$\infty$$
$$D_0 - 2 \cdot(1 : -1 : 0)$$ $$z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0.110794$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$2$$ Mordell-Weil rank: $$2$$ 2-Selmer rank: $$2$$ Regulator: $$0.016682$$ Real period: $$19.31974$$ Tamagawa product: $$1$$ Torsion order: $$1$$ Leading coefficient: $$0.322300$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$7549$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 148 T + 7549 T^{2} )$$

## 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 $$\Q$$.