# Properties

 Label 644.a.659456.1 Conductor $644$ Discriminant $659456$ Mordell-Weil group $$\Z/{2}\Z$$ Sato-Tate group $\mathrm{SU}(2)\times\mathrm{SU}(2)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q \times \Q$$ $$\End(J) \otimes \Q$$ $$\Q \times \Q$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Related objects

Show commands for: SageMath / Magma

## Simplified equation

 $y^2 + (x^2 + x)y = -3x^6 - 13x^5 + 4x^4 + 51x^3 + 4x^2 - 13x - 3$ (homogenize, simplify) $y^2 + (x^2z + xz^2)y = -3x^6 - 13x^5z + 4x^4z^2 + 51x^3z^3 + 4x^2z^4 - 13xz^5 - 3z^6$ (dehomogenize, simplify) $y^2 = -12x^6 - 52x^5 + 17x^4 + 206x^3 + 17x^2 - 52x - 12$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-3, -13, 4, 51, 4, -13, -3]), R([0, 1, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-3, -13, 4, 51, 4, -13, -3], R![0, 1, 1]);

sage: X = HyperellipticCurve(R([-12, -52, 17, 206, 17, -52, -12]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$644$$ $$=$$ $$2^{2} \cdot 7 \cdot 23$$ magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(644,2),R![1, 2, 1]>*])); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$659456$$ $$=$$ $$2^{12} \cdot 7 \cdot 23$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$161796$$ $$=$$ $$2^{2} \cdot 3 \cdot 97 \cdot 139$$ $$I_4$$ $$=$$ $$1070662305$$ $$=$$ $$3 \cdot 5 \cdot 23 \cdot 1487 \cdot 2087$$ $$I_6$$ $$=$$ $$46065265919409$$ $$=$$ $$3 \cdot 11 \cdot 1395917149073$$ $$I_{10}$$ $$=$$ $$84410368$$ $$=$$ $$2^{19} \cdot 7 \cdot 23$$ $$J_2$$ $$=$$ $$40449$$ $$=$$ $$3 \cdot 97 \cdot 139$$ $$J_4$$ $$=$$ $$23560804$$ $$=$$ $$2^{2} \cdot 199 \cdot 29599$$ $$J_6$$ $$=$$ $$14638854160$$ $$=$$ $$2^{4} \cdot 5 \cdot 7 \cdot 23 \cdot 1136557$$ $$J_8$$ $$=$$ $$9253881697856$$ $$=$$ $$2^{6} \cdot 54101 \cdot 2672629$$ $$J_{10}$$ $$=$$ $$659456$$ $$=$$ $$2^{12} \cdot 7 \cdot 23$$ $$g_1$$ $$=$$ $$108277681088425330677249/659456$$ $$g_2$$ $$=$$ $$389810454818831018649/164864$$ $$g_3$$ $$=$$ $$9297727292338785/256$$

sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]

magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);

## Automorphism group

 $$\mathrm{Aut}(X)$$ $$\simeq$$ $C_2^2$ magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$$C_2^2$magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);

## Rational points

 All points: $$(-3 : -3 : 1),\, (-1 : 3 : 3)$$ All points: $$(-3 : -3 : 1),\, (-1 : 3 : 3)$$ All points: $$(-3 : 0 : 1),\, (-1 : 0 : 3)$$

magma: [C![-3,-3,1],C![-1,3,3]]; // minimal model

magma: [C![-3,0,1],C![-1,0,3]]; // simplified model

Number of rational Weierstrass points: $$2$$

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: $$\Z/{2}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(-3 : -3 : 1) + (-1 : 3 : 3) - D_\infty$$ $$(x + 3z) (3x + z)$$ $$=$$ $$0,$$ $$6y$$ $$=$$ $$7xz^2 + 3z^3$$ $$0$$ $$2$$
Generator $D_0$ Height Order
$$(-3 : -3 : 1) + (-1 : 3 : 3) - D_\infty$$ $$(x + 3z) (3x + z)$$ $$=$$ $$0,$$ $$6y$$ $$=$$ $$7xz^2 + 3z^3$$ $$0$$ $$2$$
Generator $D_0$ Height Order
$$D_0 - D_\infty$$ $$(x + 3z) (3x + z)$$ $$=$$ $$0,$$ $$6y$$ $$=$$ $$x^2z + 15xz^2 + 6z^3$$ $$0$$ $$2$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$1$$ Regulator: $$1$$ Real period: $$0.872984$$ Tamagawa product: $$1$$ Torsion order: $$2$$ Leading coefficient: $$0.218246$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$2$$ $$12$$ $$1$$ $$( 1 + T )^{2}$$
$$7$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 4 T + 7 T^{2} )$$
$$23$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 23 T^{2} )$$

## Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $\mathrm{SU}(2)\times\mathrm{SU}(2)$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{SU}(2)\times\mathrm{SU}(2)$$

## Decomposition of the Jacobian

Splits over $$\Q$$

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
Elliptic curve isogeny class 46.a
Elliptic curve isogeny class 14.a

## Endomorphisms of the Jacobian

Of $$\GL_2$$-type over $$\Q$$

Endomorphism ring over $$\Q$$:

 $$\End (J_{})$$ $$\simeq$$ an order of index $$2$$ in $$\Z \times \Z$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\times$$ $$\Q$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R \times \R$$

All $$\overline{\Q}$$-endomorphisms of the Jacobian are defined over $$\Q$$.