# Properties

 Label 644.a.659456.1 Conductor 644 Discriminant 659456 Sato-Tate group $G_{3,3}$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q \times \Q$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Related objects

Show commands for: Magma / SageMath

## Minimal equation

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

$y^2 + (x^2 + x)y = -3x^6 - 13x^5 + 4x^4 + 51x^3 + 4x^2 - 13x - 3$

## Invariants

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

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$323592$$ = $$2^{3} \cdot 3 \cdot 97 \cdot 139$$ $$I_4$$ = $$4282649220$$ = $$2^{2} \cdot 3 \cdot 5 \cdot 23 \cdot 1487 \cdot 2087$$ $$I_6$$ = $$368522127355272$$ = $$2^{3} \cdot 3 \cdot 11 \cdot 1395917149073$$ $$I_{10}$$ = $$2701131776$$ = $$2^{24} \cdot 7 \cdot 23$$ $$J_2$$ = $$40449$$ = $$3 \cdot 97 \cdot 139$$ $$J_4$$ = $$23560804$$ = $$2^{2} \cdot 199 \cdot 29599$$ $$J_6$$ = $$14638854160$$ = $$2^{4} \cdot 5 \cdot 7 \cdot 23 \cdot 1136557$$ $$J_8$$ = $$9253881697856$$ = $$2^{6} \cdot 54101 \cdot 2672629$$ $$J_{10}$$ = $$659456$$ = $$2^{12} \cdot 7 \cdot 23$$ $$g_1$$ = $$108277681088425330677249/659456$$ $$g_2$$ = $$389810454818831018649/164864$$ $$g_3$$ = $$9297727292338785/256$$
Alternative geometric invariants: G2

## Automorphism group

 magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X)$$ $$\simeq$$ $$V_4$$ (GAP id : [4,2]) magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$ $$V_4$$ (GAP id : [4,2])

## Rational points

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.

magma: [C![-3,-3,1],C![-1,3,3]];

All rational points: (-3 : -3 : 1), (-1 : 3 : 3)

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

Number of rational Weierstrass points: $$2$$

## Invariants of the Jacobian:

Analytic rank: $$0$$

magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1); 2-Selmer rank: $$1$$ magma: HasSquareSha(Jacobian(C)); Order of Ш*: square Tamagawa numbers: 1 (p = 2), 1 (p = 7), 1 (p = 23) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

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

### Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $G_{3,3}$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{SU}(2)\times\mathrm{SU}(2)$$

### Decomposition

Splits over $$\Q$$

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
Elliptic curve 46.a1
Elliptic curve 14.a1

### Endomorphisms

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 $$\Q$$.