# Properties

 Label 799680.b.799680.1 Conductor 799680 Discriminant -799680 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![446, 0, 336, 0, 84, 0, 7], R![1, 0, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([446, 0, 336, 0, 84, 0, 7]), R([1, 0, 1]))

$y^2 + (x^2 + 1)y = 7x^6 + 84x^4 + 336x^2 + 446$

## Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$799680$$ = $$2^{6} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 17$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$-799680$$ = $$-1 \cdot 2^{6} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 17$$

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$-19252832$$ = $$-1 \cdot 2^{5} \cdot 601651$$ $$I_4$$ = $$228856000$$ = $$2^{6} \cdot 5^{3} \cdot 28607$$ $$I_6$$ = $$-1467556505638912$$ = $$-1 \cdot 2^{11} \cdot 13 \cdot 163 \cdot 338169101$$ $$I_{10}$$ = $$-3275489280$$ = $$-1 \cdot 2^{18} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 17$$ $$J_2$$ = $$-2406604$$ = $$-1 \cdot 2^{2} \cdot 601651$$ $$J_4$$ = $$241320233284$$ = $$2^{2} \cdot 13 \cdot 4640773717$$ $$J_6$$ = $$-32263933356762240$$ = $$-1 \cdot 2^{7} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 17 \cdot 89 \cdot 151 \cdot 1501081$$ $$J_8$$ = $$4852764019968313102076$$ = $$2^{2} \cdot 41 \cdot 47 \cdot 127 \cdot 632911 \cdot 7832512601$$ $$J_{10}$$ = $$-799680$$ = $$-1 \cdot 2^{6} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 17$$ $$g_1$$ = $$1261372031256529020641523732016/12495$$ $$g_2$$ = $$52556648780600635811581759084/12495$$ $$g_3$$ = $$233673974755154692792288$$
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: [];

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: 2.30612468166 Real period: 2.0358049513911927833065139047 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: $$\Z/{6}\Z$$

2-torsion field: splitting field of $$x^{8} - 4 x^{7} + 90 x^{6} - 200 x^{5} + 2123 x^{4} - 2480 x^{3} + 14610 x^{2} + 3668 x + 10306$$ 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 57120.bd1
Elliptic curve 14.a4

### 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 $$\Q$$.
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