Genus 2 curve 2916.b.11664.1

• Label: 2916.b.11664.1
• Conductor: $2916$
• Discriminant: $-11664$
• Mordell-Weil group: \(\Z/{3}\Z \oplus \Z/{3}\Z\)
• 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

Minimal equation

$y^2 + y = x^6$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(2916\)	\(=\)	\( 2^{2} \cdot 3^{6} \)
Discriminant:	\( \Delta \)	\(=\)	\(-11664\)	\(=\)	\( - 2^{4} \cdot 3^{6} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(40\)	\(=\)	\( 2^{3} \cdot 5 \)
\( I_4 \)	\(=\)	\(45\)	\(=\)	\( 3^{2} \cdot 5 \)
\( I_6 \)	\(=\)	\(555\)	\(=\)	\( 3 \cdot 5 \cdot 37 \)
\( I_{10} \)	\(=\)	\(6\)	\(=\)	\( 2 \cdot 3 \)

Automorphism group

\(\mathrm{Aut}(X)\)	\(\simeq\)	$C_2^2$
\(\mathrm{Aut}(X_{\overline{\Q}})\)	\(\simeq\)	$C_3:D_4$

Rational points

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

Number of rational Weierstrass points: \(0\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: \(\Z/{3}\Z \oplus \Z/{3}\Z\)

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

2-torsion field: 6.0.186624.1

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
\(2\)	\(2\)	\(4\)	\(3\)	\(1 + 2 T^{2}\)
\(3\)	\(6\)	\(6\)	\(1\)	\(1\)

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?
\(2\)	2.60.2	no
\(3\)	3.17280.1	yes

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

Additional information

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 $C_3:D_4$ with GAP id $\langle 24,8\rangle$).