# Properties

 Label 4624.d.295936.1 Conductor 4624 Discriminant 295936 Mordell-Weil group $$\Z/{13}\Z$$ 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

 $y^2 + (x^3 + x^2)y = x^3 + x^2 - 2x - 1$ (homogenize, simplify) $y^2 + (x^3 + x^2z)y = x^3z^3 + x^2z^4 - 2xz^5 - z^6$ (dehomogenize, simplify) $y^2 = x^6 + 2x^5 + x^4 + 4x^3 + 4x^2 - 8x - 4$ (minimize, homogenize)

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

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

sage: X = HyperellipticCurve(R([-4, -8, 4, 4, 1, 2, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$4624$$ $$=$$ $$2^{4} \cdot 17^{2}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$295936$$ $$=$$ $$2^{10} \cdot 17^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$352$$ $$=$$ $$2^{5} \cdot 11$$ $$I_4$$ $$=$$ $$-20672$$ $$=$$ $$- 2^{6} \cdot 17 \cdot 19$$ $$I_6$$ $$=$$ $$-9256704$$ $$=$$ $$- 2^{8} \cdot 3 \cdot 17 \cdot 709$$ $$I_{10}$$ $$=$$ $$1212153856$$ $$=$$ $$2^{22} \cdot 17^{2}$$ $$J_2$$ $$=$$ $$44$$ $$=$$ $$2^{2} \cdot 11$$ $$J_4$$ $$=$$ $$296$$ $$=$$ $$2^{3} \cdot 37$$ $$J_6$$ $$=$$ $$13636$$ $$=$$ $$2^{2} \cdot 7 \cdot 487$$ $$J_8$$ $$=$$ $$128092$$ $$=$$ $$2^{2} \cdot 31 \cdot 1033$$ $$J_{10}$$ $$=$$ $$295936$$ $$=$$ $$2^{10} \cdot 17^{2}$$ $$g_1$$ $$=$$ $$161051/289$$ $$g_2$$ $$=$$ $$49247/578$$ $$g_3$$ $$=$$ $$412489/4624$$

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

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

Number of rational Weierstrass points: $$1$$

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: $$\Z/{13}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(-1 : 1 : 1) - (1 : -1 : 0)$$ $$z (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$z^3$$ $$0$$ $$13$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$0$$ Regulator: $$1$$ Real period: $$12.76618$$ Tamagawa product: $$13$$ Torsion order: $$13$$ Leading coefficient: $$0.982014$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$4$$ $$10$$ $$13$$ $$1 - T$$
$$17$$ $$2$$ $$2$$ $$1$$ $$1 + 8 T + 17 T^{2}$$

## 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 $$\Q$$.