# Properties

 Label 2916.b.11664.1 Conductor $2916$ Discriminant $-11664$ Mordell-Weil group $$\Z/{3}\Z \times \Z/{3}\Z$$ Sato-Tate group $D_{3,2}$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\mathrm{M}_2(\C)$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathrm{M}_2(\mathsf{CM})$$ $$\End(J) \otimes \Q$$ $$\Q \times \Q$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Related objects

Show commands for: SageMath / Magma

## Simplified equation

 $y^2 + y = x^6$ (homogenize, simplify) $y^2 + z^3y = x^6$ (dehomogenize, simplify) $y^2 = 4x^6 + 1$ (minimize, homogenize)

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

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

sage: X = HyperellipticCurve(R([1, 0, 0, 0, 0, 0, 4]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$2916$$ $$=$$ $$2^{2} \cdot 3^{6}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-11664$$ $$=$$ $$- 2^{4} \cdot 3^{6}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$40$$ $$=$$ $$2^{3} \cdot 5$$ $$I_4$$ $$=$$ $$45$$ $$=$$ $$3^{2} \cdot 5$$ $$I_6$$ $$=$$ $$555$$ $$=$$ $$3 \cdot 5 \cdot 37$$ $$I_{10}$$ $$=$$ $$6$$ $$=$$ $$2 \cdot 3$$ $$J_2$$ $$=$$ $$120$$ $$=$$ $$2^{3} \cdot 3 \cdot 5$$ $$J_4$$ $$=$$ $$330$$ $$=$$ $$2 \cdot 3 \cdot 5 \cdot 11$$ $$J_6$$ $$=$$ $$-320$$ $$=$$ $$- 2^{6} \cdot 5$$ $$J_8$$ $$=$$ $$-36825$$ $$=$$ $$- 3 \cdot 5^{2} \cdot 491$$ $$J_{10}$$ $$=$$ $$11664$$ $$=$$ $$2^{4} \cdot 3^{6}$$ $$g_1$$ $$=$$ $$6400000/3$$ $$g_2$$ $$=$$ $$440000/9$$ $$g_3$$ $$=$$ $$-32000/81$$

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

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

magma: [C![0,-1,1],C![0,1,1],C![1,-2,0],C![1,2,0]]; // 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: $$\Z/{3}\Z \times \Z/{3}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

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

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$0$$ Regulator: $$1$$ Real period: $$17.69503$$ Tamagawa product: $$3$$ Torsion order: $$9$$ Leading coefficient: $$0.655371$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$2$$ $$4$$ $$3$$ $$1 + 2 T^{2}$$
$$3$$ $$6$$ $$6$$ $$1$$ $$1$$

## Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $D_{3,2}$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{U}(1)$$

## 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 108.a
Elliptic curve isogeny class 27.a

## 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 $$K = \Q (a) \simeq$$ 6.0.34992.1 with defining polynomial $$x^{6} - 3 x^{5} + 5 x^{3} - 3 x + 1$$

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 $$F \simeq$$ $$\Q(\sqrt{-3})$$ with generator $$2 a^{5} - 5 a^{4} - 2 a^{3} + 8 a^{2} + 4 a - 3$$ 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: $C_3$
Not of $$\GL_2$$-type, not simple

Over subfield $$F \simeq$$ 3.1.108.1 with generator $$2 a^{5} - 5 a^{4} - 3 a^{3} + 10 a^{2} + 5 a - 5$$ with minimal polynomial $$x^{3} - 2$$:

 $$\End (J_{F})$$ $$\simeq$$ 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 $$F \simeq$$ 3.1.108.1 with generator $$-a^{2} + a + 1$$ with minimal polynomial $$x^{3} - 2$$:

 $$\End (J_{F})$$ $$\simeq$$ 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 $$F \simeq$$ 3.1.108.1 with generator $$-2 a^{5} + 5 a^{4} + 3 a^{3} - 9 a^{2} - 6 a + 4$$ with minimal polynomial $$x^{3} - 2$$:

 $$\End (J_{F})$$ $$\simeq$$ 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

The conductor 2916 of the Jacobian of this curve is the smallest known for a genus 2 Jacobian that is $\overline\Q$-isogenous to the square of an elliptic curve, or equivalently whose Sato-Tate group has identity component isomorphic to $\mathrm U(1)$.
It is also the smallest known example of a genus 2 curve whose geometric automorphism group has order 24 (the group $D_3:C_4$ with GAP id $\langle 24,8\rangle$).