# Properties

 Label 1253.a.1253.1 Conductor 1253 Discriminant -1253 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![-330, 15, 43, -33, 0, 2, -1], R![1, 0, 1, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-330, 15, 43, -33, 0, 2, -1]), R([1, 0, 1, 1]))

$y^2 + (x^3 + x^2 + 1)y = -x^6 + 2x^5 - 33x^3 + 43x^2 + 15x - 330$

## Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$1253$$ = $$7 \cdot 179$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$-1253$$ = $$-1 \cdot 7 \cdot 179$$

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$-827064$$ = $$-1 \cdot 2^{3} \cdot 3^{3} \cdot 7 \cdot 547$$ $$I_4$$ = $$37524148644$$ = $$2^{2} \cdot 3 \cdot 3127012387$$ $$I_6$$ = $$-7994893805035992$$ = $$-1 \cdot 2^{3} \cdot 3^{3} \cdot 739 \cdot 24019 \cdot 2085257$$ $$I_{10}$$ = $$-5132288$$ = $$-1 \cdot 2^{12} \cdot 7 \cdot 179$$ $$J_2$$ = $$-103383$$ = $$-1 \cdot 3^{3} \cdot 7 \cdot 547$$ $$J_4$$ = $$54458647$$ = $$54458647$$ $$J_6$$ = $$97243994481$$ = $$3 \cdot 29 \cdot 79 \cdot 179 \cdot 79043$$ $$J_8$$ = $$-3254780028624958$$ = $$-1 \cdot 2 \cdot 47 \cdot 34625319453457$$ $$J_{10}$$ = $$-1253$$ = $$-1 \cdot 7 \cdot 179$$ $$g_1$$ = $$1687126365978608485162449/179$$ $$g_2$$ = $$8596391751971448839127/179$$ $$g_3$$ = $$-829487756384515053$$
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 except over $\R$.

magma: [];

There are no rational points.

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*: twice a square Regulator: 1.0 Real period: 0.20746393501932595902618920284 Tamagawa numbers: 1 (p = 7), 1 (p = 179) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

Torsion: $$\mathrm{trivial}$$

### 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 $$\Q$$.