# Properties

 Label 1184.a.606208.1 Conductor 1184 Discriminant 606208 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

## Minimal equation

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([]))

$y^2 = 2x^5 + x^4 - 8x^3 - 8x^2 - 2x$

## Invariants

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

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$5632$$ = $$2^{9} \cdot 11$$ $$I_4$$ = $$507904$$ = $$2^{14} \cdot 31$$ $$I_6$$ = $$980353024$$ = $$2^{16} \cdot 7 \cdot 2137$$ $$I_{10}$$ = $$2483027968$$ = $$2^{26} \cdot 37$$ $$J_2$$ = $$704$$ = $$2^{6} \cdot 11$$ $$J_4$$ = $$15360$$ = $$2^{10} \cdot 3 \cdot 5$$ $$J_6$$ = $$140288$$ = $$2^{10} \cdot 137$$ $$J_8$$ = $$-34291712$$ = $$-1 \cdot 2^{14} \cdot 7 \cdot 13 \cdot 23$$ $$J_{10}$$ = $$606208$$ = $$2^{14} \cdot 37$$ $$g_1$$ = $$10554638336/37$$ $$g_2$$ = $$327106560/37$$ $$g_3$$ = $$4243712/37$$
Alternative geometric invariants: G2

## Automorphism group

 magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X)$$ $$\simeq$$ $$C_2$$ (GAP id : [2,1]) magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$ $$C_2$$ (GAP id : [2,1])

## Rational points

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.

magma: [C![-1,-1,1],C![-1,0,2],C![-1,1,1],C![0,0,1],C![1,0,0]];

All rational points: (-1 : -1 : 1), (-1 : 0 : 2), (-1 : 1 : 1), (0 : 0 : 1), (1 : 0 : 0)

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

Number of rational Weierstrass points: $$3$$

## Invariants of the Jacobian:

Analytic rank: $$0$$

magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1); 2-Selmer rank: $$2$$ magma: HasSquareSha(Jacobian(C)); Order of Ш*: square Tamagawa numbers: 8 (p = 2), 1 (p = 37) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

Torsion: $$\Z/{2}\Z \times \Z/{8}\Z$$

### Sato-Tate group

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

### Decomposition

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

### Endomorphisms

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 $$\Q$$.