# Properties

 Label 1184.a.606208.2 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![2, -2, 6, -4, 5, -2, 1], R![]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([2, -2, 6, -4, 5, -2, 1]), R([]))

$y^2 = x^6 - 2x^5 + 5x^4 - 4x^3 + 6x^2 - 2x + 2$

## 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); $$\Delta$$ = $$-606208$$ = $$-1 \cdot 2^{14} \cdot 37$$

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$-11264$$ = $$-1 \cdot 2^{10} \cdot 11$$ $$I_4$$ = $$323584$$ = $$2^{12} \cdot 79$$ $$I_6$$ = $$-1122041856$$ = $$-1 \cdot 2^{16} \cdot 3 \cdot 13 \cdot 439$$ $$I_{10}$$ = $$-2483027968$$ = $$-1 \cdot 2^{26} \cdot 37$$ $$J_2$$ = $$-1408$$ = $$-1 \cdot 2^{7} \cdot 11$$ $$J_4$$ = $$79232$$ = $$2^{7} \cdot 619$$ $$J_6$$ = $$-5831680$$ = $$-1 \cdot 2^{10} \cdot 5 \cdot 17 \cdot 67$$ $$J_8$$ = $$483323904$$ = $$2^{12} \cdot 3^{2} \cdot 7 \cdot 1873$$ $$J_{10}$$ = $$-606208$$ = $$-1 \cdot 2^{14} \cdot 37$$ $$g_1$$ = $$337748426752/37$$ $$g_2$$ = $$13498597376/37$$ $$g_3$$ = $$705633280/37$$
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,-1,0],C![1,1,0]];

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

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

Number of rational Weierstrass points: $$0$$

## Invariants of the Jacobian:

Analytic rank: $$0$$

magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1); 2-Selmer rank: $$1$$ magma: HasSquareSha(Jacobian(C)); Order of Ш*: square Tamagawa numbers: 4 (p = 2), 1 (p = 37) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

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

### 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 $$\Q$$.