# Properties

 Label 100152.a.600912.1 Conductor 100152 Discriminant -600912 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

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, 0, -4, -6, -2, 1], R![1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, 0, -4, -6, -2, 1]), R([1]))

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, 0, -4, -6, -2, 1], R![1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, 4, 0, -16, -24, -8, 4]))

 $y^2 + y = x^6 - 2x^5 - 6x^4 - 4x^3 + x$ (homogenize, simplify) $y^2 + z^3y = x^6 - 2x^5z - 6x^4z^2 - 4x^3z^3 + xz^5$ (dehomogenize, simplify) $y^2 = 4x^6 - 8x^5 - 24x^4 - 16x^3 + 4x + 1$ (minimize, homogenize)

## Invariants

 $$N$$ = $$100152$$ = $$2^{3} \cdot 3^{2} \cdot 13 \cdot 107$$ magma: Conductor(LSeries(C)); Factorization($1); $$\Delta$$ = $$-600912$$ = $$- 2^{4} \cdot 3^{3} \cdot 13 \cdot 107$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$-704$$ = $$- 2^{6} \cdot 11$$ $$I_4$$ = $$-79040$$ = $$- 2^{6} \cdot 5 \cdot 13 \cdot 19$$ $$I_6$$ = $$7366144$$ = $$2^{9} \cdot 14387$$ $$I_{10}$$ = $$-2461335552$$ = $$- 2^{16} \cdot 3^{3} \cdot 13 \cdot 107$$ $$J_2$$ = $$-88$$ = $$- 2^{3} \cdot 11$$ $$J_4$$ = $$1146$$ = $$2 \cdot 3 \cdot 191$$ $$J_6$$ = $$5760$$ = $$2^{7} \cdot 3^{2} \cdot 5$$ $$J_8$$ = $$-455049$$ = $$- 3^{2} \cdot 7 \cdot 31 \cdot 233$$ $$J_{10}$$ = $$-600912$$ = $$- 2^{4} \cdot 3^{3} \cdot 13 \cdot 107$$ $$g_1$$ = $$329832448/37557$$ $$g_2$$ = $$16270144/12519$$ $$g_3$$ = $$-309760/4173$$

## Automorphism group

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

## Rational points

magma: [C![-1,-45,4],C![-1,-19,4],C![-1,-14,3],C![-1,-13,3],C![-1,-1,1],C![-1,0,1],C![0,-1,1],C![0,0,1],C![1,-1,0],C![1,1,0]];

Known points
$$(0 : 0 : 1)$$ $$(-1 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(1 : -1 : 0)$$ $$(1 : 1 : 0)$$ $$(-1 : -1 : 1)$$
$$(-1 : -13 : 3)$$ $$(-1 : -14 : 3)$$ $$(-1 : -19 : 4)$$ $$(-1 : -45 : 4)$$

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

Number of rational Weierstrass points: $$0$$

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.

## Mordell-Weil group of the Jacobian:

magma: MordellWeilGroupGenus2(Jacobian(C));

Group structure: $$\Z \times \Z$$

Generator Height Order
$$2x^2 + 2xz + z^2$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$xz^2 - z^3$$ $$0.354578$$ $$\infty$$
$$x (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0.058359$$ $$\infty$$

## BSD invariants

 Analytic rank: $$2$$   (upper bound) Mordell-Weil rank: $$2$$ 2-Selmer rank: $$2$$ Regulator: $$0.020121$$ Real period: $$11.02077$$ Tamagawa product: $$6$$ Torsion order: $$1$$ Leading coefficient: $$1.330500$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$2$$ $$4$$ $$3$$ $$2$$ $$1 + 2 T + 2 T^{2}$$
$$3$$ $$3$$ $$2$$ $$3$$ $$( 1 - T )( 1 + T )$$
$$13$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 5 T + 13 T^{2} )$$
$$107$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 12 T + 107 T^{2} )$$

## 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 $$\Q$$.