# Properties

 Label 799680.b.799680.1 Conductor 799680 Discriminant -799680 Mordell-Weil group $$\Z \times \Z/{6}\Z$$ 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

## Simplified equation

 $y^2 + (x^2 + 1)y = 7x^6 + 84x^4 + 336x^2 + 446$ (homogenize, simplify) $y^2 + (x^2z + z^3)y = 7x^6 + 84x^4z^2 + 336x^2z^4 + 446z^6$ (dehomogenize, simplify) $y^2 = 28x^6 + 337x^4 + 1346x^2 + 1785$ (minimize, homogenize)

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]));

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([1785, 0, 1346, 0, 337, 0, 28]))

## Invariants

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

### G2 invariants

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

magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);

sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]

## Automorphism group

 $$\mathrm{Aut}(X)$$ $$\simeq$$ $C_2^2$ magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$$C_2^2$magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);

## Rational points

No rational points are known for this curve.

magma: [];

Number of rational Weierstrass points: $$0$$

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

This curve is locally solvable everywhere.

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);

## Mordell-Weil group of the Jacobian:

Group structure: $$\Z \times \Z/{6}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - D_\infty$$ $$x^2 + 5z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$xz^2 + 2z^3$$ $$2.306124$$ $$\infty$$
$$D_0 - D_\infty$$ $$x^2 + 4z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$2z^3$$ $$0$$ $$6$$

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 $D_4\times C_2$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$6$$ Regulator: $$2.306124$$ Real period: $$2.035804$$ Tamagawa product: $$1$$ Torsion order: $$6$$ Leading coefficient: $$2.086586$$ Analytic order of Ш: $$16$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$2$$ $$6$$ $$6$$ $$1$$ $$1 + T$$
$$3$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 2 T + 3 T^{2} )$$
$$5$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 5 T^{2} )$$
$$7$$ $$2$$ $$2$$ $$1$$ $$( 1 - T )^{2}$$
$$17$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 - 6 T + 17 T^{2} )$$

## Sato-Tate group

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

## Decomposition of the Jacobian

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 the Jacobian

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 $$\Q$$.