# Properties

 Label 114240.d.114240.1 Conductor 114240 Discriminant 114240 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![-9, 0, 103, 0, -315, 0, 51], R![1, 0, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-9, 0, 103, 0, -315, 0, 51]), R([1, 0, 1]))

$y^2 + (x^2 + 1)y = 51x^6 - 315x^4 + 103x^2 - 9$

## Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$114240$$ = $$2^{6} \cdot 3 \cdot 5 \cdot 7 \cdot 17$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$114240$$ = $$2^{6} \cdot 3 \cdot 5 \cdot 7 \cdot 17$$

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$10053216$$ = $$2^{5} \cdot 3^{2} \cdot 67 \cdot 521$$ $$I_4$$ = $$3341086802112$$ = $$2^{6} \cdot 3 \cdot 29 \cdot 600051509$$ $$I_6$$ = $$9160538337022040064$$ = $$2^{11} \cdot 3^{2} \cdot 151 \cdot 3847 \cdot 855557891$$ $$I_{10}$$ = $$467927040$$ = $$2^{18} \cdot 3 \cdot 5 \cdot 7 \cdot 17$$ $$J_2$$ = $$1256652$$ = $$2^{2} \cdot 3^{2} \cdot 67 \cdot 521$$ $$J_4$$ = $$30995939524$$ = $$2^{2} \cdot 11 \cdot 704453171$$ $$J_6$$ = $$838652756430720$$ = $$2^{7} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 17 \cdot 7789 \cdot 42841$$ $$J_8$$ = $$23286599174677950716$$ = $$2^{2} \cdot 11 \cdot 1217233 \cdot 434790126733$$ $$J_{10}$$ = $$114240$$ = $$2^{6} \cdot 3 \cdot 5 \cdot 7 \cdot 17$$ $$g_1$$ = $$16322019979578992366124730896/595$$ $$g_2$$ = $$320367650677941067851565476/595$$ $$g_3$$ = $$11592952003636922822112$$
Alternative geometric invariants: G2

## Automorphism group

 magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X)$$ $$\simeq$$ $$V_4$$ (GAP id : [4,2]) magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$ $$V_4$$ (GAP id : [4,2])

## 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: [];

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: $$6$$ magma: HasSquareSha(Jacobian(C)); Order of Ш*: square Regulator: 1.0 Real period: 0.16518202509712685856962351073 Tamagawa numbers: 1 (p = 2), 1 (p = 3), 1 (p = 5), 1 (p = 7), 1 (p = 17) 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 1120.j1
Elliptic curve 102.c1

### Endomorphisms

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

Endomorphism ring over $$\Q$$:
 $$\End (J_{})$$ $$\simeq$$ an order of index $$2$$ 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 $$\Q$$.