# Properties

 Label 294.a.8232.1 Conductor $294$ Discriminant $8232$ Mordell-Weil group $$\Z/{12}\Z$$ Sato-Tate group $\mathrm{SU}(2)\times\mathrm{SU}(2)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q \times \Q$$ $$\End(J) \otimes \Q$$ $$\Q \times \Q$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Related objects

Show commands: Magma / SageMath

## Simplified equation

 $y^2 + (x^3 + 1)y = -2x^4 + 4x^2 - 9x - 14$ (homogenize, simplify) $y^2 + (x^3 + z^3)y = -2x^4z^2 + 4x^2z^4 - 9xz^5 - 14z^6$ (dehomogenize, simplify) $y^2 = x^6 - 8x^4 + 2x^3 + 16x^2 - 36x - 55$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-14, -9, 4, 0, -2]), R([1, 0, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-14, -9, 4, 0, -2], R![1, 0, 0, 1]);

sage: X = HyperellipticCurve(R([-55, -36, 16, 2, -8, 0, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$294$$ $$=$$ $$2 \cdot 3 \cdot 7^{2}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$8232$$ $$=$$ $$2^{3} \cdot 3 \cdot 7^{3}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$7636$$ $$=$$ $$2^{2} \cdot 23 \cdot 83$$ $$I_4$$ $$=$$ $$11785$$ $$=$$ $$5 \cdot 2357$$ $$I_6$$ $$=$$ $$29745701$$ $$=$$ $$239 \cdot 124459$$ $$I_{10}$$ $$=$$ $$1053696$$ $$=$$ $$2^{10} \cdot 3 \cdot 7^{3}$$ $$J_2$$ $$=$$ $$1909$$ $$=$$ $$23 \cdot 83$$ $$J_4$$ $$=$$ $$151354$$ $$=$$ $$2 \cdot 7 \cdot 19 \cdot 569$$ $$J_6$$ $$=$$ $$15951264$$ $$=$$ $$2^{5} \cdot 3 \cdot 7^{2} \cdot 3391$$ $$J_8$$ $$=$$ $$1885732415$$ $$=$$ $$5 \cdot 7^{2} \cdot 7696867$$ $$J_{10}$$ $$=$$ $$8232$$ $$=$$ $$2^{3} \cdot 3 \cdot 7^{3}$$ $$g_1$$ $$=$$ $$25353016669288549/8232$$ $$g_2$$ $$=$$ $$75211396489919/588$$ $$g_3$$ $$=$$ $$49431027484/7$$

sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]

magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);

## Automorphism group

 $$\mathrm{Aut}(X)$$ $$\simeq$$ $C_2^2$ magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$$C_2^2$magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);

## Rational points

All points: $$(1 : 0 : 0),\, (1 : -1 : 0),\, (-2 : 3 : 1),\, (-2 : 4 : 1)$$
All points: $$(1 : 0 : 0),\, (1 : -1 : 0),\, (-2 : 3 : 1),\, (-2 : 4 : 1)$$
All points: $$(1 : -1 : 0),\, (1 : 1 : 0),\, (-2 : -1 : 1),\, (-2 : 1 : 1)$$

magma: [C![-2,3,1],C![-2,4,1],C![1,-1,0],C![1,0,0]]; // minimal model

magma: [C![-2,-1,1],C![-2,1,1],C![1,-1,0],C![1,1,0]]; // simplified model

Number of rational Weierstrass points: $$0$$

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

This curve is locally solvable everywhere.

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);

## Mordell-Weil group of the Jacobian

Group structure: $$\Z/{12}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3$$ $$0$$ $$12$$
Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3$$ $$0$$ $$12$$
Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : 1 : 0)$$ $$z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3 + z^3$$ $$0$$ $$12$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$1$$ Regulator: $$1$$ Real period: $$7.150511$$ Tamagawa product: $$3$$ Torsion order: $$12$$ Leading coefficient: $$0.148968$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$1$$ $$3$$ $$1$$ $$( 1 + T )( 1 + T + 2 T^{2} )$$
$$3$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 2 T + 3 T^{2} )$$
$$7$$ $$2$$ $$3$$ $$3$$ $$( 1 - T )( 1 + T )$$

## Galois representations

For primes $\ell \ge 5$ the Galois representation data has not been computed for this curve since it is not generic.

For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not all of $\GSp(4,\F_\ell)$.

Prime $$\ell$$ mod-$$\ell$$ image Is torsion prime?
$$2$$ 2.45.1 yes
$$3$$ 3.2160.20 yes

## Sato-Tate group

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

## Decomposition of the Jacobian

Splits over $$\Q$$

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
Elliptic curve isogeny class 14.a
Elliptic curve isogeny class 21.a

magma: HeuristicDecompositionFactors(C);

## Endomorphisms of the Jacobian

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 $$\Q$$.

magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

magma: HeuristicIsGL2(C); HeuristicEndomorphismDescription(C); HeuristicEndomorphismFieldOfDefinition(C);

magma: HeuristicIsGL2(C : Geometric := true); HeuristicEndomorphismDescription(C : Geometric := true); HeuristicEndomorphismLatticeDescription(C);