# Properties

 Label 20736.k.373248.1 Conductor $20736$ Discriminant $-373248$ Mordell-Weil group $$\Z \oplus \Z/{6}\Z$$ Sato-Tate group $D_{2,1}$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\mathrm{M}_2(\C)$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathrm{M}_2(\mathsf{CM})$$ $$\End(J) \otimes \Q$$ $$\Q \times \Q$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Related objects

Show commands: Magma / SageMath

## Simplified equation

 $y^2 + x^3y = 2$ (homogenize, simplify) $y^2 + x^3y = 2z^6$ (dehomogenize, simplify) $y^2 = x^6 + 8$ (homogenize, minimize)

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

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

sage: X = HyperellipticCurve(R([8, 0, 0, 0, 0, 0, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$20736$$ $$=$$ $$2^{8} \cdot 3^{4}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-373248$$ $$=$$ $$- 2^{9} \cdot 3^{6}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$40$$ $$=$$ $$2^{3} \cdot 5$$ $$I_4$$ $$=$$ $$45$$ $$=$$ $$3^{2} \cdot 5$$ $$I_6$$ $$=$$ $$555$$ $$=$$ $$3 \cdot 5 \cdot 37$$ $$I_{10}$$ $$=$$ $$6$$ $$=$$ $$2 \cdot 3$$ $$J_2$$ $$=$$ $$240$$ $$=$$ $$2^{4} \cdot 3 \cdot 5$$ $$J_4$$ $$=$$ $$1320$$ $$=$$ $$2^{3} \cdot 3 \cdot 5 \cdot 11$$ $$J_6$$ $$=$$ $$-2560$$ $$=$$ $$- 2^{9} \cdot 5$$ $$J_8$$ $$=$$ $$-589200$$ $$=$$ $$- 2^{4} \cdot 3 \cdot 5^{2} \cdot 491$$ $$J_{10}$$ $$=$$ $$373248$$ $$=$$ $$2^{9} \cdot 3^{6}$$ $$g_1$$ $$=$$ $$6400000/3$$ $$g_2$$ $$=$$ $$440000/9$$ $$g_3$$ $$=$$ $$-32000/81$$

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_3:D_4$magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);

## Rational points

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

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

magma: [C![-1,-3,1],C![-1,3,1],C![1,-3,1],C![1,-1,0],C![1,1,0],C![1,3,1]]; // 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: $$\Z \oplus \Z/{6}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(-1 : -1 : 1) - (1 : -1 : 0)$$ $$z (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0.326617$$ $$\infty$$
$$(-1 : 2 : 1) + (1 : -2 : 1) - (1 : -1 : 0) - (1 : 0 : 0)$$ $$(x - z) (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-2xz^2$$ $$0$$ $$6$$
Generator $D_0$ Height Order
$$(-1 : -1 : 1) - (1 : -1 : 0)$$ $$z (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0.326617$$ $$\infty$$
$$(-1 : 2 : 1) + (1 : -2 : 1) - (1 : -1 : 0) - (1 : 0 : 0)$$ $$(x - z) (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-2xz^2$$ $$0$$ $$6$$
Generator $D_0$ Height Order
$$(-1 : -3 : 1) - (1 : -1 : 0)$$ $$z (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^3 - 2z^3$$ $$0.326617$$ $$\infty$$
$$(-1 : 3 : 1) + (1 : -3 : 1) - (1 : -1 : 0) - (1 : 1 : 0)$$ $$(x - z) (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^3 - 4xz^2$$ $$0$$ $$6$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$2$$ Regulator: $$0.326617$$ Real period: $$12.51227$$ Tamagawa product: $$12$$ Torsion order: $$6$$ Leading coefficient: $$1.362242$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$8$$ $$9$$ $$3$$ $$1$$
$$3$$ $$4$$ $$6$$ $$4$$ $$1$$

## 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?
$$2$$ 2.180.4 yes
$$3$$ 3.8640.8 yes

## Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $D_{2,1}$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{U}(1)$$

## 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 576.e
Elliptic curve isogeny class 36.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 $$\Z \times \Z$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\times$$ $$\Q$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R \times \R$$

Smallest field over which all endomorphisms are defined:
Galois number field $$K = \Q (a) \simeq$$ $$\Q(\sqrt{2}, \sqrt{-3})$$ with defining polynomial $$x^{4} + 2 x^{2} + 4$$

Not of $$\GL_2$$-type over $$\overline{\Q}$$

Endomorphism ring over $$\overline{\Q}$$:

 $$\End (J_{\overline{\Q}})$$ $$\simeq$$ a non-Eichler order of index $$16$$ in a maximal order of $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\simeq$$ $$\mathrm{M}_2($$$$\Q(\sqrt{-3})$$$$)$$ $$\End (J_{\overline{\Q}}) \otimes \R$$ $$\simeq$$ $$\mathrm{M}_2 (\C)$$

### Remainder of the endomorphism lattice by field

Over subfield $$F \simeq$$ $$\Q(\sqrt{2})$$ with generator $$\frac{1}{2} a^{3}$$ with minimal polynomial $$x^{2} - 2$$:

 $$\End (J_{F})$$ $$\simeq$$ a non-Eichler order of index $$4$$ in a maximal order of $$\End (J_{F}) \otimes \Q$$ $$\End (J_{F}) \otimes \Q$$ $$\simeq$$ $$\mathrm{M}_2($$$$\Q$$$$)$$ $$\End (J_{F}) \otimes \R$$ $$\simeq$$ $$\mathrm{M}_2 (\R)$$
Sato Tate group: C_{2,1}
Not of $$\GL_2$$-type, not simple

Over subfield $$F \simeq$$ $$\Q(\sqrt{-6})$$ with generator $$-\frac{1}{2} a^{3} - 2 a$$ with minimal polynomial $$x^{2} + 6$$:

 $$\End (J_{F})$$ $$\simeq$$ a non-Eichler order of index $$12$$ in a maximal order of $$\End (J_{F}) \otimes \Q$$ $$\End (J_{F}) \otimes \Q$$ $$\simeq$$ $$\mathrm{M}_2($$$$\Q$$$$)$$ $$\End (J_{F}) \otimes \R$$ $$\simeq$$ $$\mathrm{M}_2 (\R)$$
Sato Tate group: C_{2,1}
Not of $$\GL_2$$-type, not simple

Over subfield $$F \simeq$$ $$\Q(\sqrt{-3})$$ with generator $$-\frac{1}{2} a^{2}$$ with minimal polynomial $$x^{2} - x + 1$$:

 $$\End (J_{F})$$ $$\simeq$$ an order of index $$4$$ in $$\Z [\frac{1 + \sqrt{-3}}{2}] \times \Z [\frac{1 + \sqrt{-3}}{2}]$$ $$\End (J_{F}) \otimes \Q$$ $$\simeq$$ $$\Q(\sqrt{-3})$$ $$\times$$ $$\Q(\sqrt{-3})$$ $$\End (J_{F}) \otimes \R$$ $$\simeq$$ $$\C \times \C$$
Sato Tate group: C_2
Not of $$\GL_2$$-type, not simple

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);