# Properties

 Label 1184.a.606208.1 Conductor 1184 Discriminant 606208 Mordell-Weil group $$\Z/{2}\Z \times \Z/{8}\Z$$ 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

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

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -2, -8, -8, 1, 2], R![]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -2, -8, -8, 1, 2]), R([]));

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([0, -2, -8, -8, 1, 2]))

## Invariants

 Conductor: $$N$$ = $$1184$$ = $$2^{5} \cdot 37$$ magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(1184,2),R![1]>*])); Factorization($1); Discriminant: $$\Delta$$ = $$606208$$ = $$2^{14} \cdot 37$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$5632$$ = $$2^{9} \cdot 11$$ $$I_4$$ = $$507904$$ = $$2^{14} \cdot 31$$ $$I_6$$ = $$980353024$$ = $$2^{16} \cdot 7 \cdot 2137$$ $$I_{10}$$ = $$2483027968$$ = $$2^{26} \cdot 37$$ $$J_2$$ = $$704$$ = $$2^{6} \cdot 11$$ $$J_4$$ = $$15360$$ = $$2^{10} \cdot 3 \cdot 5$$ $$J_6$$ = $$140288$$ = $$2^{10} \cdot 137$$ $$J_8$$ = $$-34291712$$ = $$- 2^{14} \cdot 7 \cdot 13 \cdot 23$$ $$J_{10}$$ = $$606208$$ = $$2^{14} \cdot 37$$ $$g_1$$ = $$10554638336/37$$ $$g_2$$ = $$327106560/37$$ $$g_3$$ = $$4243712/37$$

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: $$(1 : 0 : 0),\, (0 : 0 : 1),\, (-1 : -1 : 1),\, (-1 : 1 : 1),\, (-1 : 0 : 2)$$

magma: [C![-1,-1,1],C![-1,0,2],C![-1,1,1],C![0,0,1],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: $$\Z/{2}\Z \times \Z/{8}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(-1 : 0 : 2) - (1 : 0 : 0)$$ $$2x + z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$2$$
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$x^2 - 3xz - 2z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$7xz^2 + 4z^3$$ $$0$$ $$8$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$2$$ Regulator: $$1$$ Real period: $$14.23204$$ Tamagawa product: $$8$$ Torsion order: $$16$$ Leading coefficient: $$0.444751$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$2$$ $$14$$ $$5$$ $$8$$ $$1$$
$$37$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 - 6 T + 37 T^{2} )$$

## 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 $$\Q$$.