# Properties

 Label 100124.a.400496.1 Conductor 100124 Discriminant -400496 Mordell-Weil group $$\Z \times \Z/{2}\Z \times \Z/{2}\Z$$ 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

 $y^2 + (x^2 + x)y = x^5 - 2x^4 + 2x^2 - 2x$ (homogenize, simplify) $y^2 + (x^2z + xz^2)y = x^5z - 2x^4z^2 + 2x^2z^4 - 2xz^5$ (dehomogenize, simplify) $y^2 = 4x^5 - 7x^4 + 2x^3 + 9x^2 - 8x$ (minimize, homogenize)

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

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

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([0, -8, 9, 2, -7, 4]))

## Invariants

 Conductor: $$N$$ = $$100124$$ = $$2^{2} \cdot 25031$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$-400496$$ = $$- 2^{4} \cdot 25031$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$-248$$ = $$- 2^{3} \cdot 31$$ $$I_4$$ = $$-60956$$ = $$- 2^{2} \cdot 7^{2} \cdot 311$$ $$I_6$$ = $$-839960$$ = $$- 2^{3} \cdot 5 \cdot 11 \cdot 23 \cdot 83$$ $$I_{10}$$ = $$-1640431616$$ = $$- 2^{16} \cdot 25031$$ $$J_2$$ = $$-31$$ = $$- 31$$ $$J_4$$ = $$675$$ = $$3^{3} \cdot 5^{2}$$ $$J_6$$ = $$6857$$ = $$6857$$ $$J_8$$ = $$-167048$$ = $$- 2^{3} \cdot 7 \cdot 19 \cdot 157$$ $$J_{10}$$ = $$-400496$$ = $$- 2^{4} \cdot 25031$$ $$g_1$$ = $$28629151/400496$$ $$g_2$$ = $$20108925/400496$$ $$g_3$$ = $$-6589577/400496$$

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

All points: $$(1 : 0 : 0),\, (0 : 0 : 1),\, (-1 : -1 : 1),\, (-1 : 1 : 1),\, (1 : -1 : 1)$$

magma: [C![-1,-1,1],C![-1,1,1],C![0,0,1],C![1,-1,1],C![1,0,0]];

Number of rational Weierstrass points: $$3$$

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: $$\Z \times \Z/{2}\Z \times \Z/{2}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(-1 : -1 : 1) - (1 : 0 : 0)$$ $$x + z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0.822181$$ $$\infty$$
$$(0 : 0 : 1) - (1 : 0 : 0)$$ $$x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$2$$
$$(1 : -1 : 1) - (1 : 0 : 0)$$ $$x - z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0$$ $$2$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$3$$ Regulator: $$0.822181$$ Real period: $$11.14761$$ Tamagawa product: $$4$$ Torsion order: $$4$$ Leading coefficient: $$2.291339$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$2$$ $$4$$ $$2$$ $$4$$ $$( 1 - T )( 1 + T )$$
$$25031$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 - 56 T + 25031 T^{2} )$$

## 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 $$\Q$$.