Genus 2 curve 12321.a.36963.1

• Label: 12321.a.36963.1
• Conductor: $12321$
• Discriminant: $-36963$
• Mordell-Weil group: \(\Z \oplus \Z\)
• Sato-Tate group: $E_6$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\mathrm{M}_2(\R)\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathrm{M}_2(\Q)\)
• \(\End(J) \otimes \Q\): \(\mathsf{CM}\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: yes

Minimal equation

$y^2 + (x^3 + x + 1)y = x^5 + 3x^4 + 4x^3 + 2x^2$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(12321\)	\(=\)	\( 3^{2} \cdot 37^{2} \)
Discriminant:	\( \Delta \)	\(=\)	\(-36963\)	\(=\)	\( - 3^{3} \cdot 37^{2} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(4\)	\(=\)	\( 2^{2} \)
\( I_4 \)	\(=\)	\(6697\)	\(=\)	\( 37 \cdot 181 \)
\( I_6 \)	\(=\)	\(85285\)	\(=\)	\( 5 \cdot 37 \cdot 461 \)
\( I_{10} \)	\(=\)	\(-4731264\)	\(=\)	\( - 2^{7} \cdot 3^{3} \cdot 37^{2} \)

Automorphism group

\(\mathrm{Aut}(X)\)	\(\simeq\)	$C_6$
\(\mathrm{Aut}(X_{\overline{\Q}})\)	\(\simeq\)	$D_6$

Rational points

Known points
\((1 : 0 : 0)\)	\((1 : -1 : 0)\)	\((0 : 0 : 1)\)	\((-1 : 0 : 1)\)	\((0 : -1 : 1)\)	\((-1 : 1 : 1)\)
\((-2 : 1 : 1)\)	\((1 : 2 : 1)\)	\((-1 : 2 : 2)\)	\((1 : -5 : 1)\)	\((-1 : -5 : 2)\)	\((-2 : 8 : 1)\)

Number of rational Weierstrass points: \(0\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: \(\Z \oplus \Z\)

Generator	$D_0$						Height	Order
\((0 : -1 : 1) - (1 : -1 : 0)\)	\(z x\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-z^3\)	\(0.098911\)	\(\infty\)
\((-1 : 0 : 1) + (0 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\)	\(x (x + z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0.098911\)	\(\infty\)

2-torsion field: 6.0.36963.1

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	\(2\)
Mordell-Weil rank:	\(2\)
2-Selmer rank:	\(2\)
Regulator:	\( 0.007337 \)
Real period:	\( 18.95274 \)
Tamagawa product:	\( 3 \)
Torsion order:	\( 1 \)
Leading coefficient:	\( 0.417205 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	Root number	L-factor	Cluster picture	Tame reduction?
\(3\)	\(2\)	\(3\)	\(3\)	\(1\)	\(1 + T + T^{2}\)		yes
\(37\)	\(2\)	\(2\)	\(1\)	\(1\)	\(1 + 10 T + 37 T^{2}\)		yes

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?
\(2\)	2.40.3	no
\(3\)	3.480.12	no

Sato-Tate group

\(\mathrm{ST}\)	\(\simeq\)	$E_6$
\(\mathrm{ST}^0\)	\(\simeq\)	\(\mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) 6.6.69343957.1 with defining polynomial:
  \(x^{6} - x^{5} - 15 x^{4} + 28 x^{3} + 15 x^{2} - 38 x - 1\)

Decomposes up to isogeny as the square of the elliptic curve isogeny class:
  \(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
  \(g_4 = \frac{27938}{11} b^{5} - \frac{102400}{11} b^{4} + \frac{41669}{11} b^{3} + \frac{90470}{11} b^{2} - \frac{45088}{11} b + \frac{110611}{11}\)
  \(g_6 = \frac{32589452}{11} b^{5} - \frac{166685962}{11} b^{4} + \frac{182285865}{11} b^{3} + \frac{203348152}{11} b^{2} - \frac{315712416}{11} b - \frac{19805878}{11}\)
   Conductor norm: 729

Endomorphisms of the Jacobian

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

Endomorphism ring over \(\Q\):

\(\End (J_{})\)	\(\simeq\)	\(\Z [\frac{1 + \sqrt{-3}}{2}]\)
\(\End (J_{}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{-3}) \)
\(\End (J_{}) \otimes \R\)	\(\simeq\)	\(\C\)

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) 6.6.69343957.1 with defining polynomial \(x^{6} - x^{5} - 15 x^{4} + 28 x^{3} + 15 x^{2} - 38 x - 1\)

Not of \(\GL_2\)-type over \(\overline{\Q}\)

Endomorphism ring over \(\overline{\Q}\):

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

Remainder of the endomorphism lattice by field

Over subfield \(F \simeq \) \(\Q(\sqrt{37}) \) with generator \(\frac{2}{11} a^{5} + \frac{1}{11} a^{4} - \frac{23}{11} a^{3} + \frac{27}{11} a^{2} - \frac{1}{11} a - \frac{28}{11}\) with minimal polynomial \(x^{2} - x - 9\):

\(\End (J_{F})\)	\(\simeq\)	\(\Z [\frac{1 + \sqrt{-3}}{2}]\)
\(\End (J_{F}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{-3}) \)
\(\End (J_{F}) \otimes \R\)	\(\simeq\)	\(\C\)

  Sato Tate group: $E_3$
  Of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) 3.3.1369.1 with generator \(-\frac{4}{11} a^{5} - \frac{2}{11} a^{4} + \frac{57}{11} a^{3} - \frac{32}{11} a^{2} - \frac{108}{11} a + \frac{45}{11}\) with minimal polynomial \(x^{3} - x^{2} - 12 x - 11\):

\(\End (J_{F})\)	\(\simeq\)	\(\Z [\frac{1 + \sqrt{-3}}{2}]\)
\(\End (J_{F}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{-3}) \)
\(\End (J_{F}) \otimes \R\)	\(\simeq\)	\(\C\)

  Sato Tate group: $E_2$
  Of \(\GL_2\)-type, simple