# Properties

 Label 24704.a.790528.1 Conductor 24704 Discriminant -790528 Mordell-Weil group $$\Z \times \Z$$ Sato-Tate group $\mathrm{USp}(4)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type no

# Related objects

Show commands for: Magma / SageMath

## Simplified equation

 $y^2 + x^3y = 6x^3 + 14x^2 + 12x + 4$ (homogenize, simplify) $y^2 + x^3y = 6x^3z^3 + 14x^2z^4 + 12xz^5 + 4z^6$ (dehomogenize, simplify) $y^2 = x^6 + 24x^3 + 56x^2 + 48x + 16$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![4, 12, 14, 6], R![0, 0, 0, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([4, 12, 14, 6]), R([0, 0, 0, 1]));

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([16, 48, 56, 24, 0, 0, 1]))

## Invariants

 Conductor: $$N$$ = $$24704$$ = $$2^{7} \cdot 193$$ magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(24704,2),R![1]>*])); Factorization($1); Discriminant: $$\Delta$$ = $$-790528$$ = $$- 2^{12} \cdot 193$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$-384$$ = $$- 2^{7} \cdot 3$$ $$I_4$$ = $$218112$$ = $$2^{10} \cdot 3 \cdot 71$$ $$I_6$$ = $$-42369024$$ = $$- 2^{15} \cdot 3 \cdot 431$$ $$I_{10}$$ = $$-3238002688$$ = $$- 2^{24} \cdot 193$$ $$J_2$$ = $$-48$$ = $$- 2^{4} \cdot 3$$ $$J_4$$ = $$-2176$$ = $$- 2^{7} \cdot 17$$ $$J_6$$ = $$43008$$ = $$2^{11} \cdot 3 \cdot 7$$ $$J_8$$ = $$-1699840$$ = $$- 2^{12} \cdot 5 \cdot 83$$ $$J_{10}$$ = $$-790528$$ = $$- 2^{12} \cdot 193$$ $$g_1$$ = $$62208/193$$ $$g_2$$ = $$-58752/193$$ $$g_3$$ = $$-24192/193$$

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

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

## Automorphism group

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

## Rational points

Known points
$$(1 : 0 : 0)$$ $$(1 : -1 : 0)$$ $$(-1 : 0 : 1)$$ $$(-1 : 1 : 1)$$ $$(0 : -2 : 1)$$ $$(0 : 2 : 1)$$
$$(-2 : 2 : 1)$$ $$(-2 : 6 : 1)$$ $$(-3 : 11 : 2)$$ $$(-3 : 16 : 2)$$ $$(-5 : 44 : 3)$$ $$(-5 : 81 : 3)$$
$$(8 : 2058 : 7)$$ $$(8 : -2570 : 7)$$

magma: [C![-5,44,3],C![-5,81,3],C![-3,11,2],C![-3,16,2],C![-2,2,1],C![-2,6,1],C![-1,0,1],C![-1,1,1],C![0,-2,1],C![0,2,1],C![1,-1,0],C![1,0,0],C![8,-2570,7],C![8,2058,7]];

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: $$\Z \times \Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(-1 : 0 : 1) - (1 : -1 : 0)$$ $$z (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0.144322$$ $$\infty$$
$$(-2 : 6 : 1) - (1 : 0 : 0)$$ $$z (x + 2z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3 - 2z^3$$ $$0.051963$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$2$$ Mordell-Weil rank: $$2$$ 2-Selmer rank: $$2$$ Regulator: $$0.007471$$ Real period: $$15.42238$$ Tamagawa product: $$8$$ Torsion order: $$1$$ Leading coefficient: $$0.921871$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$2$$ $$12$$ $$7$$ $$8$$ $$1$$
$$193$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 - 17 T + 193 T^{2} )$$

## Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $\mathrm{USp}(4)$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{USp}(4)$$

## Decomposition of the Jacobian

Simple over $$\overline{\Q}$$

## Endomorphisms of the Jacobian

Not of $$\GL_2$$-type over $$\Q$$

Endomorphism ring over $$\Q$$:

 $$\End (J_{})$$ $$\simeq$$ $$\Z$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R$$

All $$\overline{\Q}$$-endomorphisms of the Jacobian are defined over $$\Q$$.