# Properties

 Label 15256.a.122048.1 Conductor 15256 Discriminant 122048 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

 $y^2 + (x + 1)y = x^6 - x^4 + 2x^2 + x$ (homogenize, simplify) $y^2 + (xz^2 + z^3)y = x^6 - x^4z^2 + 2x^2z^4 + xz^5$ (dehomogenize, simplify) $y^2 = 4x^6 - 4x^4 + 9x^2 + 6x + 1$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, 2, 0, -1, 0, 1], R![1, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, 2, 0, -1, 0, 1]), R([1, 1]));

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([1, 6, 9, 0, -4, 0, 4]))

## Invariants

 Conductor: $$N$$ = $$15256$$ = $$2^{3} \cdot 1907$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$122048$$ = $$2^{6} \cdot 1907$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$-384$$ = $$- 2^{7} \cdot 3$$ $$I_4$$ = $$67776$$ = $$2^{6} \cdot 3 \cdot 353$$ $$I_6$$ = $$-13663488$$ = $$- 2^{8} \cdot 3 \cdot 17791$$ $$I_{10}$$ = $$499908608$$ = $$2^{18} \cdot 1907$$ $$J_2$$ = $$-48$$ = $$- 2^{4} \cdot 3$$ $$J_4$$ = $$-610$$ = $$- 2 \cdot 5 \cdot 61$$ $$J_6$$ = $$14052$$ = $$2^{2} \cdot 3 \cdot 1171$$ $$J_8$$ = $$-261649$$ = $$- 19 \cdot 47 \cdot 293$$ $$J_{10}$$ = $$122048$$ = $$2^{6} \cdot 1907$$ $$g_1$$ = $$-3981312/1907$$ $$g_2$$ = $$1054080/1907$$ $$g_3$$ = $$505872/1907$$

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
$$(1 : -1 : 0)$$ $$(1 : 1 : 0)$$ $$(0 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(-1 : -1 : 1)$$ $$(-1 : 1 : 1)$$
$$(1 : 1 : 1)$$ $$(-1 : -1 : 2)$$ $$(1 : -3 : 1)$$ $$(-1 : -3 : 2)$$ $$(-1 : -17 : 4)$$ $$(2 : 17 : 3)$$
$$(-1 : -31 : 4)$$ $$(2 : -62 : 3)$$

magma: [C![-1,-31,4],C![-1,-17,4],C![-1,-3,2],C![-1,-1,1],C![-1,-1,2],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-3,1],C![1,-1,0],C![1,1,0],C![1,1,1],C![2,-62,3],C![2,17,3]];

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
$$2 \cdot(0 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)$$ $$x^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$xz^2$$ $$0.264086$$ $$\infty$$
$$(-1 : -1 : 1) - (1 : -1 : 0)$$ $$z (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^3$$ $$0.021732$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$2$$ Mordell-Weil rank: $$2$$ 2-Selmer rank: $$2$$ Regulator: $$0.005663$$ Real period: $$17.79322$$ Tamagawa product: $$6$$ Torsion order: $$1$$ Leading coefficient: $$0.604627$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$2$$ $$6$$ $$3$$ $$6$$ $$1 + T$$
$$1907$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 - 29 T + 1907 T^{2} )$$

## 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 $$\Q$$.