# Properties

 Label 100130.a.200260.1 Conductor 100130 Discriminant -200260 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, 1, 1, 0, 6, 5, 1], R![1, 1, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, 1, 0, 6, 5, 1]), R([1, 1, 1]))

$y^2 + (x^2 + x + 1)y = x^6 + 5x^5 + 6x^4 + x^2 + x$

## Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$100130$$ = $$2 \cdot 5 \cdot 17 \cdot 19 \cdot 31$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$-200260$$ = $$-1 \cdot 2^{2} \cdot 5 \cdot 17 \cdot 19 \cdot 31$$

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$1064$$ = $$2^{3} \cdot 7 \cdot 19$$ $$I_4$$ = $$872740$$ = $$2^{2} \cdot 5 \cdot 11 \cdot 3967$$ $$I_6$$ = $$-8682968$$ = $$-1 \cdot 2^{3} \cdot 7 \cdot 47 \cdot 3299$$ $$I_{10}$$ = $$-820264960$$ = $$-1 \cdot 2^{14} \cdot 5 \cdot 17 \cdot 19 \cdot 31$$ $$J_2$$ = $$133$$ = $$7 \cdot 19$$ $$J_4$$ = $$-8354$$ = $$-1 \cdot 2 \cdot 4177$$ $$J_6$$ = $$356384$$ = $$2^{5} \cdot 7 \cdot 37 \cdot 43$$ $$J_8$$ = $$-5597561$$ = $$-1 \cdot 5597561$$ $$J_{10}$$ = $$-200260$$ = $$-1 \cdot 2^{2} \cdot 5 \cdot 17 \cdot 19 \cdot 31$$ $$g_1$$ = $$-2190305047/10540$$ $$g_2$$ = $$517208671/5270$$ $$g_3$$ = $$-82948376/2635$$
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,-2,1],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-1,0],C![1,1,0],C![7,-839,3],C![7,602,3]];

Known rational points: (-1 : -2 : 1), (-1 : 1 : 1), (0 : -1 : 1), (0 : 0 : 1), (1 : -1 : 0), (1 : 1 : 0), (7 : -839 : 3), (7 : 602 : 3)

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

Number of rational Weierstrass points: $$0$$

## Invariants of the Jacobian:

Analytic rank*: $$2$$

magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1); 2-Selmer rank: $$3$$ magma: HasSquareSha(Jacobian(C)); Order of Ш*: square Tamagawa numbers: 2 (p = 2), 1 (p = 5), 1 (p = 17), 1 (p = 19), 1 (p = 31) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

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

2-torsion field: splitting field of $$x^{6} - 20 x^{4} - 26 x^{3} - 255 x^{2} - 1730 x - 1706$$ with Galois group $S_4\times C_2$

### 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 $$\Q$$.