# Properties

 Label 1055.a.1055.1 Conductor 1055 Discriminant -1055 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![-1, -1, 1, 0, -1], R![1, 0, 0, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, -1, 1, 0, -1]), R([1, 0, 0, 1]))

$y^2 + (x^3 + 1)y = -x^4 + x^2 - x - 1$

## Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$1055$$ = $$5 \cdot 211$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$-1055$$ = $$-1 \cdot 5 \cdot 211$$

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$1000$$ = $$2^{3} \cdot 5^{3}$$ $$I_4$$ = $$-12092$$ = $$-1 \cdot 2^{2} \cdot 3023$$ $$I_6$$ = $$-4201016$$ = $$-1 \cdot 2^{3} \cdot 525127$$ $$I_{10}$$ = $$-4321280$$ = $$-1 \cdot 2^{12} \cdot 5 \cdot 211$$ $$J_2$$ = $$125$$ = $$5^{3}$$ $$J_4$$ = $$777$$ = $$3 \cdot 7 \cdot 37$$ $$J_6$$ = $$7441$$ = $$7 \cdot 1063$$ $$J_8$$ = $$81599$$ = $$7 \cdot 11657$$ $$J_{10}$$ = $$-1055$$ = $$-1 \cdot 5 \cdot 211$$ $$g_1$$ = $$-6103515625/211$$ $$g_2$$ = $$-303515625/211$$ $$g_3$$ = $$-23253125/211$$
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![-1,0,1],C![1,-1,0],C![1,0,0]];

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

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

Number of rational Weierstrass points: $$1$$

## 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: 1 (p = 5), 1 (p = 211) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

Torsion: $$\Z/{6}\Z$$

### 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 $$\Q$$.