# Properties

 Label 672.a.172032.1 Conductor 672 Discriminant 172032 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![56, 0, -75, 0, -16, 0, -1], R![0, 1, 0, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([56, 0, -75, 0, -16, 0, -1]), R([0, 1, 0, 1]))

$y^2 + (x^3 + x)y = -x^6 - 16x^4 - 75x^2 + 56$

## Invariants

 magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(672,2),R![1, 1]>*])); Factorization($1); $$N$$ = $$672$$ = $$2^{5} \cdot 3 \cdot 7$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$172032$$ = $$2^{13} \cdot 3 \cdot 7$$

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$-135328$$ = $$-1 \cdot 2^{5} \cdot 4229$$ $$I_4$$ = $$9671540800$$ = $$2^{6} \cdot 5^{2} \cdot 6044713$$ $$I_6$$ = $$-119230681068032$$ = $$-1 \cdot 2^{9} \cdot 397 \cdot 586580413$$ $$I_{10}$$ = $$704643072$$ = $$2^{25} \cdot 3 \cdot 7$$ $$J_2$$ = $$-16916$$ = $$-1 \cdot 2^{2} \cdot 4229$$ $$J_4$$ = $$-88822256$$ = $$-1 \cdot 2^{4} \cdot 103 \cdot 53897$$ $$J_6$$ = $$-277597802496$$ = $$-1 \cdot 2^{12} \cdot 3 \cdot 7 \cdot 3227281$$ $$J_8$$ = $$-798387183476800$$ = $$-1 \cdot 2^{6} \cdot 5^{2} \cdot 107 \cdot 991 \cdot 4705829$$ $$J_{10}$$ = $$172032$$ = $$2^{13} \cdot 3 \cdot 7$$ $$g_1$$ = $$-1352659309173012149/168$$ $$g_2$$ = $$419870026410625699/168$$ $$g_3$$ = $$-461744933079368$$
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 except over $\Q_{2}$.

magma: [];

There are no rational points.

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*: twice a square Tamagawa numbers: 2 (p = 2), 1 (p = 3), 1 (p = 7) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

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

### 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.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 $$\Q$$.