# Properties

 Label 676.b.17576.1 Conductor $676$ Discriminant $-17576$ Mordell-Weil group $$\Z/{3}\Z \oplus \Z/{3}\Z$$ Sato-Tate group $E_1$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\mathrm{M}_2(\R)$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathrm{M}_2(\Q)$$ $$\End(J) \otimes \Q$$ $$\mathrm{M}_2(\Q)$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type no

# Related objects

Show commands: Magma / SageMath

## Simplified equation

 $y^2 + (x^2 + x)y = -x^6 + 3x^5 - 6x^4 + 6x^3 - 6x^2 + 3x - 1$ (homogenize, simplify) $y^2 + (x^2z + xz^2)y = -x^6 + 3x^5z - 6x^4z^2 + 6x^3z^3 - 6x^2z^4 + 3xz^5 - z^6$ (dehomogenize, simplify) $y^2 = -4x^6 + 12x^5 - 23x^4 + 26x^3 - 23x^2 + 12x - 4$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, 3, -6, 6, -6, 3, -1]), R([0, 1, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, 3, -6, 6, -6, 3, -1], R![0, 1, 1]);

sage: X = HyperellipticCurve(R([-4, 12, -23, 26, -23, 12, -4]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$676$$ $$=$$ $$2^{2} \cdot 13^{2}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-17576$$ $$=$$ $$- 2^{3} \cdot 13^{3}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$1244$$ $$=$$ $$2^{2} \cdot 311$$ $$I_4$$ $$=$$ $$1249$$ $$=$$ $$1249$$ $$I_6$$ $$=$$ $$129167$$ $$=$$ $$37 \cdot 3491$$ $$I_{10}$$ $$=$$ $$2249728$$ $$=$$ $$2^{10} \cdot 13^{3}$$ $$J_2$$ $$=$$ $$311$$ $$=$$ $$311$$ $$J_4$$ $$=$$ $$3978$$ $$=$$ $$2 \cdot 3^{2} \cdot 13 \cdot 17$$ $$J_6$$ $$=$$ $$72332$$ $$=$$ $$2^{2} \cdot 13^{2} \cdot 107$$ $$J_8$$ $$=$$ $$1667692$$ $$=$$ $$2^{2} \cdot 13^{2} \cdot 2467$$ $$J_{10}$$ $$=$$ $$17576$$ $$=$$ $$2^{3} \cdot 13^{3}$$ $$g_1$$ $$=$$ $$2909390022551/17576$$ $$g_2$$ $$=$$ $$4602275343/676$$ $$g_3$$ $$=$$ $$10349147/26$$

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

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

## Automorphism group

 $$\mathrm{Aut}(X)$$ $$\simeq$$ $D_6$ magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$$D_6$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 $\R$, $\Q_{2}$, and $\Q_{11}$.

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: $$\Z/{3}\Z \oplus \Z/{3}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

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

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$0$$ Regulator: $$1$$ Real period: $$7.177120$$ Tamagawa product: $$3$$ Torsion order: $$9$$ Leading coefficient: $$0.265819$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$2$$ $$3$$ $$1$$ $$( 1 + T )^{2}$$
$$13$$ $$2$$ $$3$$ $$3$$ $$( 1 - T )^{2}$$

## 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?
$$2$$ 2.120.4 no
$$3$$ 3.17280.1 yes

## Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $E_1$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{SU}(2)$$

## Decomposition of the Jacobian

Splits over $$\Q$$

Decomposes up to isogeny as the square of the elliptic curve isogeny class:
Elliptic curve isogeny class 26.a

magma: HeuristicDecompositionFactors(C);

## Endomorphisms of the Jacobian

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

Endomorphism ring over $$\Q$$:

 $$\End (J_{})$$ $$\simeq$$ an Eichler order of index $$3$$ in a maximal order of $$\End (J_{}) \otimes \Q$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\mathrm{M}_2($$$$\Q$$$$)$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\mathrm{M}_2 (\R)$$

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);