# Properties

 Label 464.a.29696.1 Conductor 464 Discriminant -29696 Mordell-Weil group $$\Z/{2}\Z \times \Z/{8}\Z$$ 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

# Learn more about

Show commands for: Magma / SageMath

## Simplified equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, -2, -4, 3, 8], R![1, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, -2, -4, 3, 8]), R([1, 1]))

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, -2, -4, 3, 8], R![1, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, 2, -7, -16, 12, 32]))

 $y^2 + (x + 1)y = 8x^5 + 3x^4 - 4x^3 - 2x^2$ (homogenize, simplify) $y^2 + (xz^2 + z^3)y = 8x^5z + 3x^4z^2 - 4x^3z^3 - 2x^2z^4$ (dehomogenize, simplify) $y^2 = 32x^5 + 12x^4 - 16x^3 - 7x^2 + 2x + 1$ (minimize, homogenize)

## Invariants

 $$N$$ = $$464$$ = $$2^{4} \cdot 29$$ magma: Conductor(LSeries(C)); Factorization($1); $$\Delta$$ = $$-29696$$ = $$- 2^{10} \cdot 29$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$5440$$ = $$2^{6} \cdot 5 \cdot 17$$ $$I_4$$ = $$-336320$$ = $$- 2^{6} \cdot 5 \cdot 1051$$ $$I_6$$ = $$-642023936$$ = $$- 2^{9} \cdot 1253953$$ $$I_{10}$$ = $$-121634816$$ = $$- 2^{22} \cdot 29$$ $$J_2$$ = $$680$$ = $$2^{3} \cdot 5 \cdot 17$$ $$J_4$$ = $$22770$$ = $$2 \cdot 3^{2} \cdot 5 \cdot 11 \cdot 23$$ $$J_6$$ = $$1180736$$ = $$2^{6} \cdot 19 \cdot 971$$ $$J_8$$ = $$71106895$$ = $$5 \cdot 223 \cdot 63773$$ $$J_{10}$$ = $$-29696$$ = $$- 2^{10} \cdot 29$$ $$g_1$$ = $$-141985700000/29$$ $$g_2$$ = $$-6991813125/29$$ $$g_3$$ = $$-533176100/29$$

## Automorphism group

 magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X)$$ $$\simeq$$$C_2$magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$ $C_2$

## Rational points

magma: [C![-1,-2,2],C![0,-1,1],C![0,0,1],C![1,-6,2],C![1,0,0]];

Points: $$(0 : 0 : 1),\, (1 : 0 : 0),\, (0 : -1 : 1),\, (-1 : -2 : 2),\, (1 : -6 : 2)$$

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));

Number of rational Weierstrass points: $$3$$

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);

This curve is locally solvable everywhere.

## Mordell-Weil group of the Jacobian:

magma: MordellWeilGroupGenus2(Jacobian(C));

Group structure: $$\Z/{2}\Z \times \Z/{8}\Z$$

Generator Height Order
$$2x - z$$ $$=$$ $$0,$$ $$4y$$ $$=$$ $$-3z^3$$ $$0$$ $$2$$
$$x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$8$$

## BSD invariants

 Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$2$$ Regulator: $$1$$ Real period: $$14.42143$$ Tamagawa product: $$4$$ Torsion order: $$16$$ Leading coefficient: $$0.225334$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$2$$ $$10$$ $$4$$ $$4$$ $$1 + T + 2 T^{2}$$
$$29$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 2 T + 29 T^{2} )$$

## 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 $$\Q$$.