Genus 2 curve 5184.a.46656.1

• Label: 5184.a.46656.1
• Conductor: $5184$
• Discriminant: $-46656$
• Mordell-Weil group: \(\Z/{6}\Z\)
• Sato-Tate group: $J(C_2)$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\mathrm{M}_2(\C)\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathrm{M}_2(\mathsf{CM})\)
• \(\End(J) \otimes \Q\): \(\mathsf{CM}\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: yes

Minimal equation

$y^2 + x^3y = x^3 + 2$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(5184\)	\(=\)	\( 2^{6} \cdot 3^{4} \)
Discriminant:	\( \Delta \)	\(=\)	\(-46656\)	\(=\)	\( - 2^{6} \cdot 3^{6} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(76\)	\(=\)	\( 2^{2} \cdot 19 \)
\( I_4 \)	\(=\)	\(252\)	\(=\)	\( 2^{2} \cdot 3^{2} \cdot 7 \)
\( I_6 \)	\(=\)	\(5160\)	\(=\)	\( 2^{3} \cdot 3 \cdot 5 \cdot 43 \)
\( I_{10} \)	\(=\)	\(24\)	\(=\)	\( 2^{3} \cdot 3 \)

Automorphism group

\(\mathrm{Aut}(X)\)	\(\simeq\)	$C_2$
\(\mathrm{Aut}(X_{\overline{\Q}})\)	\(\simeq\)	$D_6$

Rational points

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

Number of rational Weierstrass points: \(0\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: \(\Z/{6}\Z\)

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

2-torsion field: \(\Q(\zeta_{12})\)

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	\(0\)
Mordell-Weil rank:	\(0\)
2-Selmer rank:	\(1\)
Regulator:	\( 1 \)
Real period:	\( 14.11020 \)
Tamagawa product:	\( 2 \)
Torsion order:	\( 6 \)
Leading coefficient:	\( 0.783900 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

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

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
\(3\)	3.8640.16	yes

Sato-Tate group

\(\mathrm{ST}\)	\(\simeq\)	$J(C_2)$
\(\mathrm{ST}^0\)	\(\simeq\)	\(\mathrm{U}(1)\)

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) \(\Q(\sqrt{2}) \) with defining polynomial:
  \(x^{2} - 2\)

Decomposes up to isogeny as the square of the elliptic curve isogeny class:
  Elliptic curve isogeny class 2.2.8.1-81.1-b

Endomorphisms of the Jacobian

Of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)	\(\simeq\)	\(\Z [\sqrt{-2}]\)
\(\End (J_{}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{-2}) \)
\(\End (J_{}) \otimes \R\)	\(\simeq\)	\(\C\)

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 maximal order of \(\End (J_{\overline{\Q}}) \otimes \Q\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)	\(\simeq\)	\(\mathrm{M}_2(\)\(\Q(\sqrt{-6}) \)\()\)
\(\End (J_{\overline{\Q}}) \otimes \R\)	\(\simeq\)	\(\mathrm{M}_2 (\C)\)

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 \(8\) 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 \(x^{2} + 6\):

\(\End (J_{F})\)	\(\simeq\)	an order of conductor of norm \(2\) in \(\End (J_{F}) \otimes \Q\)
\(\End (J_{F}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{-2}, \sqrt{3})\) (CM)
\(\End (J_{F}) \otimes \R\)	\(\simeq\)	\(\C \times \C\)

  Sato Tate group: $C_2$
  Not of \(\GL_2\)-type, 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 Eichler order of index \(3\) in a maximal order of \(\End (J_{F}) \otimes \Q\)
\(\End (J_{F}) \otimes \Q \)	\(\simeq\)	the quaternion algebra over \(\Q\) of discriminant 2
\(\End (J_{F}) \otimes \R\)	\(\simeq\)	\(\H\)

  Sato Tate group: $J(C_1)$
  Not of \(\GL_2\)-type, simple