# Properties

 Label 720.a.6480.1 Conductor 720 Discriminant -6480 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![5, 0, 7, 0, 2], R![0, 1, 0, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([5, 0, 7, 0, 2]), R([0, 1, 0, 1]))

$y^2 + (x^3 + x)y = 2x^4 + 7x^2 + 5$

## Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$720$$ = $$2^{4} \cdot 3^{2} \cdot 5$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$-6480$$ = $$-1 \cdot 2^{4} \cdot 3^{4} \cdot 5$$

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$-9440$$ = $$-1 \cdot 2^{5} \cdot 5 \cdot 59$$ $$I_4$$ = $$191872$$ = $$2^{7} \cdot 1499$$ $$I_6$$ = $$-579060480$$ = $$-1 \cdot 2^{8} \cdot 3 \cdot 5 \cdot 150797$$ $$I_{10}$$ = $$-26542080$$ = $$-1 \cdot 2^{16} \cdot 3^{4} \cdot 5$$ $$J_2$$ = $$-1180$$ = $$-1 \cdot 2^{2} \cdot 5 \cdot 59$$ $$J_4$$ = $$56018$$ = $$2 \cdot 37 \cdot 757$$ $$J_6$$ = $$-3453120$$ = $$-1 \cdot 2^{6} \cdot 3^{2} \cdot 5 \cdot 11 \cdot 109$$ $$J_8$$ = $$234166319$$ = $$3299 \cdot 70981$$ $$J_{10}$$ = $$-6480$$ = $$-1 \cdot 2^{4} \cdot 3^{4} \cdot 5$$ $$g_1$$ = $$28596971960000/81$$ $$g_2$$ = $$1150492082200/81$$ $$g_3$$ = $$6677950400/9$$
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![1,-1,0],C![1,0,0]];

All rational points: (1 : -1 : 0), (1 : 0 : 0)

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

Number of rational Weierstrass points: $$0$$

## Invariants of the Jacobian:

Analytic rank: $$0$$

magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1); 2-Selmer rank: $$2$$ magma: HasSquareSha(Jacobian(C)); Order of Ш*: square Tamagawa numbers: 1 (p = 2), 2 (p = 3), 1 (p = 5) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

Torsion: $$\Z/{2}\Z \times \Z/{4}\Z$$

### 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 48.a4
Elliptic curve 15.a6

### 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 $$\Q$$.