# Properties

 Label 1083.b.87723.1 Conductor $1083$ Discriminant $-87723$ Mordell-Weil group $$\Z/{15}\Z$$ 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

 $y^2 + y = -x^6 - 3x^5 - 8x^4 - 11x^3 - 14x^2 - 9x - 6$ (homogenize, simplify) $y^2 + z^3y = -x^6 - 3x^5z - 8x^4z^2 - 11x^3z^3 - 14x^2z^4 - 9xz^5 - 6z^6$ (dehomogenize, simplify) $y^2 = -4x^6 - 12x^5 - 32x^4 - 44x^3 - 56x^2 - 36x - 23$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-6, -9, -14, -11, -8, -3, -1]), R([1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-6, -9, -14, -11, -8, -3, -1], R![1]);

sage: X = HyperellipticCurve(R([-23, -36, -56, -44, -32, -12, -4]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$1083$$ $$=$$ $$3 \cdot 19^{2}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-87723$$ $$=$$ $$- 3^{5} \cdot 19^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$5464$$ $$=$$ $$2^{3} \cdot 683$$ $$I_4$$ $$=$$ $$8692$$ $$=$$ $$2^{2} \cdot 41 \cdot 53$$ $$I_6$$ $$=$$ $$15768656$$ $$=$$ $$2^{4} \cdot 17 \cdot 57973$$ $$I_{10}$$ $$=$$ $$350892$$ $$=$$ $$2^{2} \cdot 3^{5} \cdot 19^{2}$$ $$J_2$$ $$=$$ $$2732$$ $$=$$ $$2^{2} \cdot 683$$ $$J_4$$ $$=$$ $$309544$$ $$=$$ $$2^{3} \cdot 38693$$ $$J_6$$ $$=$$ $$46549080$$ $$=$$ $$2^{3} \cdot 3^{5} \cdot 5 \cdot 4789$$ $$J_8$$ $$=$$ $$7838649656$$ $$=$$ $$2^{3} \cdot 29 \cdot 107 \cdot 337 \cdot 937$$ $$J_{10}$$ $$=$$ $$87723$$ $$=$$ $$3^{5} \cdot 19^{2}$$ $$g_1$$ $$=$$ $$152196082896530432/87723$$ $$g_2$$ $$=$$ $$6311963449851392/87723$$ $$g_3$$ $$=$$ $$1429770125440/361$$

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

This curve has no rational points.
This curve has no rational points.
This curve has no rational points.

magma: []; // minimal model

magma: []; // simplified model

Number of rational Weierstrass points: $$0$$

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

This curve is locally solvable except over $\R$.

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: $$\Z/{15}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - D_\infty$$ $$x^2 + 2z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$xz^2$$ $$0$$ $$15$$
Generator $D_0$ Height Order
$$D_0 - D_\infty$$ $$x^2 + 2z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$xz^2$$ $$0$$ $$15$$
Generator $D_0$ Height Order
$$D_0 - D_\infty$$ $$x^2 + 2z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$2xz^2 + z^3$$ $$0$$ $$15$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$1$$ Regulator: $$1$$ Real period: $$5.981340$$ Tamagawa product: $$5$$ Torsion order: $$15$$ Leading coefficient: $$0.265837$$ Analytic order of Ш: $$2$$   (rounded) Order of Ш: twice a square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$3$$ $$1$$ $$5$$ $$5$$ $$( 1 - T )( 1 + 2 T + 3 T^{2} )$$
$$19$$ $$2$$ $$2$$ $$1$$ $$( 1 - T )( 1 + T )$$

## 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?
$$2$$ 2.15.2 no
$$3$$ 3.720.4 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 19.a
Elliptic curve isogeny class 57.b

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);