# Properties

 Label 456960.c.913920.1 Conductor 456960 Discriminant -913920 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![-714, 0, -413, 0, -79, 0, -5], R![0, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-714, 0, -413, 0, -79, 0, -5]), R([0, 1]))

$y^2 + xy = -5x^6 - 79x^4 - 413x^2 - 714$

## Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$456960$$ = $$2^{8} \cdot 3 \cdot 5 \cdot 7 \cdot 17$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$-913920$$ = $$-1 \cdot 2^{9} \cdot 3 \cdot 5 \cdot 7 \cdot 17$$

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$-22056256$$ = $$-1 \cdot 2^{6} \cdot 344629$$ $$I_4$$ = $$932935552$$ = $$2^{7} \cdot 7288559$$ $$I_6$$ = $$-6849525368597504$$ = $$-1 \cdot 2^{10} \cdot 37 \cdot 180783503183$$ $$I_{10}$$ = $$-3743416320$$ = $$-1 \cdot 2^{21} \cdot 3 \cdot 5 \cdot 7 \cdot 17$$ $$J_2$$ = $$-2757032$$ = $$-1 \cdot 2^{3} \cdot 344629$$ $$J_4$$ = $$316708008964$$ = $$2^{2} \cdot 563 \cdot 140634107$$ $$J_6$$ = $$-48506712556968960$$ = $$-1 \cdot 2^{10} \cdot 3 \cdot 5 \cdot 7 \cdot 17 \cdot 859 \cdot 3253 \cdot 9497$$ $$J_8$$ = $$8357648948106035343356$$ = $$2^{2} \cdot 19 \cdot 41 \cdot 2682172319674594141$$ $$J_{10}$$ = $$-913920$$ = $$-1 \cdot 2^{9} \cdot 3 \cdot 5 \cdot 7 \cdot 17$$ $$g_1$$ = $$311127982838559560387716745536/1785$$ $$g_2$$ = $$12963268178085404526178374196/1785$$ $$g_3$$ = $$403438438743571080790912$$
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 $\R$ and $\Q_{2}$.

magma: [];

There are no rational points.

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

Number of rational Weierstrass points: $$0$$

## Invariants of the Jacobian:

Analytic rank: $$1$$

magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1); 2-Selmer rank: $$6$$ magma: HasSquareSha(Jacobian(C)); Order of Ш*: square Regulator: 0.519810465616 Real period: 1.5874538464207562876384391100 Tamagawa numbers: 1 (p = 2), 1 (p = 3), 1 (p = 5), 1 (p = 7), 1 (p = 17) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

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

2-torsion field: splitting field of $$x^{8} - 4 x^{7} + 66 x^{6} - 184 x^{5} + 1197 x^{4} - 2092 x^{3} + 4616 x^{2} - 3600 x + 3394$$ with Galois group $D_4\times C_2$

### 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 714.a1
Elliptic curve 640.h1

### 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 $$\Q$$.