Genus 2 curve 232324.a.464648.1

• Label: 232324.a.464648.1
• Conductor: $232324$
• Discriminant: $464648$
• 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

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

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(232324\)	\(=\)	\( 2^{2} \cdot 241^{2} \)
Discriminant:	\( \Delta \)	\(=\)	\(464648\)	\(=\)	\( 2^{3} \cdot 241^{2} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(1060\)	\(=\)	\( 2^{2} \cdot 5 \cdot 53 \)
\( I_4 \)	\(=\)	\(34945\)	\(=\)	\( 5 \cdot 29 \cdot 241 \)
\( I_6 \)	\(=\)	\(11024545\)	\(=\)	\( 5 \cdot 7 \cdot 241 \cdot 1307 \)
\( I_{10} \)	\(=\)	\(59474944\)	\(=\)	\( 2^{10} \cdot 241^{2} \)

Automorphism group

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

Rational points

Known points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (1 : -1 : 1),\, (1 : -2 : 1)\)

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
\((0 : -1 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\)	\(x (x - z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-z^3\)	\(0.235975\)	\(\infty\)
\((1 : -2 : 1) - (1 : -1 : 0)\)	\(z (x - z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-2z^3\)	\(0.235975\)	\(\infty\)

2-torsion field: 6.6.29737472.1

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	\(2\)
Mordell-Weil rank:	\(2\)
2-Selmer rank:	\(2\)
Regulator:	\( 0.041763 \)
Real period:	\( 15.98161 \)
Tamagawa product:	\( 3 \)
Torsion order:	\( 1 \)
Leading coefficient:	\( 2.002331 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(2\)	\(3\)	\(3\)	\(1 + T + T^{2}\)
\(241\)	\(2\)	\(2\)	\(1\)	\(1 - 31 T + 241 T^{2}\)

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 \(\Q (b) \simeq \) 6.6.812990017201.1 with defining polynomial:
  \(x^{6} - x^{5} - 100 x^{4} + 49 x^{3} + 2434 x^{2} - 580 x - 5875\)

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
  \(g_4 = \frac{17605023}{1504000} b^{5} + \frac{5430163}{752000} b^{4} - \frac{862065981}{752000} b^{3} - \frac{2025658079}{1504000} b^{2} + \frac{7656381061}{300800} b + \frac{9751421}{256}\)
  \(g_6 = -\frac{57988589186481}{18800000000} b^{5} - \frac{22939009858561}{9400000000} b^{4} + \frac{2863967957975407}{9400000000} b^{3} + \frac{7359068439191913}{18800000000} b^{2} - \frac{25744937066105467}{3760000000} b - \frac{32981557023387}{3200000}\)
   Conductor norm: 64

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 \(K = \Q (a) \simeq \) 6.6.812990017201.1 with defining polynomial \(x^{6} - x^{5} - 100 x^{4} + 49 x^{3} + 2434 x^{2} - 580 x - 5875\)

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 \(F \simeq \) \(\Q(\sqrt{241}) \) with generator \(\frac{1}{1175} a^{5} - \frac{18}{1175} a^{4} - \frac{29}{1175} a^{3} + \frac{1247}{1175} a^{2} - \frac{8}{47} a - 10\) with minimal polynomial \(x^{2} - x - 60\):

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

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

Over subfield \(F \simeq \) 3.3.58081.1 with generator \(\frac{1}{3760} a^{5} - \frac{9}{1880} a^{4} + \frac{103}{1880} a^{3} + \frac{307}{3760} a^{2} - \frac{2813}{752} a + \frac{71}{16}\) with minimal polynomial \(x^{3} - x^{2} - 80 x - 125\):

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

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