# Properties

 Label 1125.a.151875.1 Conductor 1125 Discriminant -151875 Sato-Tate group $G_{3,3}$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q \times \Q$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Related objects

Show commands for: Magma / SageMath

## Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 3, 22, 55, 50, 15], R![0, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 3, 22, 55, 50, 15]), R([0, 1]))

$y^2 + xy = 15x^5 + 50x^4 + 55x^3 + 22x^2 + 3x$

## Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$1125$$ = $$3^{2} \cdot 5^{3}$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$-151875$$ = $$-1 \cdot 3^{5} \cdot 5^{4}$$

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$34400$$ = $$2^{5} \cdot 5^{2} \cdot 43$$ $$I_4$$ = $$9793600$$ = $$2^{6} \cdot 5^{2} \cdot 6121$$ $$I_6$$ = $$99603070400$$ = $$2^{6} \cdot 5^{2} \cdot 62251919$$ $$I_{10}$$ = $$-622080000$$ = $$-1 \cdot 2^{12} \cdot 3^{5} \cdot 5^{4}$$ $$J_2$$ = $$4300$$ = $$2^{2} \cdot 5^{2} \cdot 43$$ $$J_4$$ = $$668400$$ = $$2^{4} \cdot 3 \cdot 5^{2} \cdot 557$$ $$J_6$$ = $$132975225$$ = $$3^{2} \cdot 5^{2} \cdot 37 \cdot 15973$$ $$J_8$$ = $$31258726875$$ = $$3^{3} \cdot 5^{4} \cdot 211 \cdot 8779$$ $$J_{10}$$ = $$-151875$$ = $$-1 \cdot 3^{5} \cdot 5^{4}$$ $$g_1$$ = $$-2352135088000000/243$$ $$g_2$$ = $$-28342655360000/81$$ $$g_3$$ = $$-437104339600/27$$
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![-4,18,3],C![0,0,1],C![1,0,0]];

All rational points: (-4 : 18 : 3), (0 : 0 : 1), (1 : 0 : 0)

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

Number of rational Weierstrass points: $$3$$

## Invariants of the Jacobian:

Analytic rank: $$0$$

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

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

### Sato-Tate group

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

### Decomposition

Splits over $$\Q$$

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
Elliptic curve 15.a2
Elliptic curve 75.c1

### Endomorphisms

Of $$\GL_2$$-type over $$\Q$$

Endomorphism ring over $$\Q$$:
 $$\End (J_{})$$ $$\simeq$$ an order of index $$3$$ in $$\Z \times \Z$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\times$$ $$\Q$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R \times \R$$

All $$\overline{\Q}$$-endomorphisms of the Jacobian are defined over $$\Q$$.