# Properties

 Label 7609.a.7609.1 Conductor 7609 Discriminant -7609 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

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

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

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

sage: X = HyperellipticCurve(R([1, -2, 5, 2, 2, 0, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$7609$$ $$=$$ $$7 \cdot 1087$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-7609$$ $$=$$ $$- 7 \cdot 1087$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$-376$$ $$=$$ $$- 2^{3} \cdot 47$$ $$I_4$$ $$=$$ $$10852$$ $$=$$ $$2^{2} \cdot 2713$$ $$I_6$$ $$=$$ $$-599704$$ $$=$$ $$- 2^{3} \cdot 7 \cdot 10709$$ $$I_{10}$$ $$=$$ $$-31166464$$ $$=$$ $$- 2^{12} \cdot 7 \cdot 1087$$ $$J_2$$ $$=$$ $$-47$$ $$=$$ $$-47$$ $$J_4$$ $$=$$ $$-21$$ $$=$$ $$- 3 \cdot 7$$ $$J_6$$ $$=$$ $$-675$$ $$=$$ $$- 3^{3} \cdot 5^{2}$$ $$J_8$$ $$=$$ $$7821$$ $$=$$ $$3^{2} \cdot 11 \cdot 79$$ $$J_{10}$$ $$=$$ $$-7609$$ $$=$$ $$- 7 \cdot 1087$$ $$g_1$$ $$=$$ $$229345007/7609$$ $$g_2$$ $$=$$ $$-311469/1087$$ $$g_3$$ $$=$$ $$1491075/7609$$

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
$$(1 : 0 : 0)$$ $$(1 : -1 : 0)$$ $$(0 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(1 : 0 : 1)$$ $$(-1 : -1 : 1)$$
$$(-1 : 2 : 1)$$ $$(1 : -3 : 1)$$ $$(3 : 1 : 2)$$ $$(3 : -48 : 2)$$

magma: [C![-1,-1,1],C![-1,2,1],C![0,-1,1],C![0,0,1],C![1,-3,1],C![1,-1,0],C![1,0,0],C![1,0,1],C![3,-48,2],C![3,1,2]];

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
$$(0 : 0 : 1) - (1 : -1 : 0)$$ $$z x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0.138985$$ $$\infty$$
$$(0 : 0 : 1) - (1 : 0 : 0)$$ $$z x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3$$ $$0.146735$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$2$$ Mordell-Weil rank: $$2$$ 2-Selmer rank: $$2$$ Regulator: $$0.017111$$ Real period: $$18.66443$$ Tamagawa product: $$1$$ Torsion order: $$1$$ Leading coefficient: $$0.319377$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$7$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 3 T + 7 T^{2} )$$
$$1087$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 - 4 T + 1087 T^{2} )$$

## 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 $$\Q$$.