# Properties

 Label 12500.a.12500.1 Conductor 12500 Discriminant 12500 Sato-Tate group $N(G_{3,3})$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathrm{RM}$$ $$\overline{\Q}$$-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, 0, 2, 0, 0, 1], R![1, 0, 0, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, 0, 2, 0, 0, 1]), R([1, 0, 0, 1]))

$y^2 + (x^3 + 1)y = x^6 + 2x^3 - x$

## Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$12500$$ = $$2^{2} \cdot 5^{5}$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$12500$$ = $$2^{2} \cdot 5^{5}$$

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$-600$$ = $$-1 \cdot 2^{3} \cdot 3 \cdot 5^{2}$$ $$I_4$$ = $$202500$$ = $$2^{2} \cdot 3^{4} \cdot 5^{4}$$ $$I_6$$ = $$-67995000$$ = $$-1 \cdot 2^{3} \cdot 3^{2} \cdot 5^{4} \cdot 1511$$ $$I_{10}$$ = $$51200000$$ = $$2^{14} \cdot 5^{5}$$ $$J_2$$ = $$-75$$ = $$-1 \cdot 3 \cdot 5^{2}$$ $$J_4$$ = $$-1875$$ = $$-1 \cdot 3 \cdot 5^{4}$$ $$J_6$$ = $$73125$$ = $$3^{2} \cdot 5^{4} \cdot 13$$ $$J_8$$ = $$-2250000$$ = $$-1 \cdot 2^{4} \cdot 3^{2} \cdot 5^{6}$$ $$J_{10}$$ = $$12500$$ = $$2^{2} \cdot 5^{5}$$ $$g_1$$ = $$-759375/4$$ $$g_2$$ = $$253125/4$$ $$g_3$$ = $$131625/4$$
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![0,-1,1],C![0,0,1]];

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

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: $$0$$ magma: HasSquareSha(Jacobian(C)); Order of Ш*: square Tamagawa numbers: 1 (p = 2), 1 (p = 5) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

Torsion: $$\Z/{5}\Z$$

### Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $N(G_{3,3})$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{SU}(2)\times\mathrm{SU}(2)$$

### Decomposition

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

### Endomorphisms

Not of $$\GL_2$$-type over $$\Q$$

Endomorphism ring over $$\Q$$:
 $$\End (J_{})$$ $$\simeq$$ $$\Z$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R$$

Smallest field over which all endomorphisms are defined:
Galois number field $$K = \Q (a) \simeq$$ $$\Q(\sqrt{5})$$ with defining polynomial $$x^{2} - x - 1$$

Of $$\GL_2$$-type over $$\overline{\Q}$$

Endomorphism ring over $$\overline{\Q}$$:
 $$\End (J_{\overline{\Q}})$$ $$\simeq$$ $$\Z [\frac{1 + \sqrt{5}}{2}]$$ $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\simeq$$ $$\Q(\sqrt{5})$$ $$\End (J_{\overline{\Q}}) \otimes \R$$ $$\simeq$$ $$\R \times \R$$