Genus 2 curve 350464.a.350464.1

• Label: 350464.a.350464.1
• Conductor: $350464$
• Discriminant: $350464$
• Mordell-Weil group: \(\Z/{2}\Z \oplus \Z/{2}\Z\)
• 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

$y^2 = x^5 - 3x^4 - 11x^3 - 8x^2 - x$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(350464\)	\(=\)	\( 2^{8} \cdot 37^{2} \)
Discriminant:	\( \Delta \)	\(=\)	\(350464\)	\(=\)	\( 2^{8} \cdot 37^{2} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(302\)	\(=\)	\( 2 \cdot 151 \)
\( I_4 \)	\(=\)	\(5032\)	\(=\)	\( 2^{3} \cdot 17 \cdot 37 \)
\( I_6 \)	\(=\)	\(388574\)	\(=\)	\( 2 \cdot 37 \cdot 59 \cdot 89 \)
\( I_{10} \)	\(=\)	\(1369\)	\(=\)	\( 37^{2} \)

Automorphism group

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

Rational points

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

Number of rational Weierstrass points: \(3\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: \(\Z/{2}\Z \oplus \Z/{2}\Z\)

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

2-torsion field: 3.3.1369.1

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	\(0\)
Mordell-Weil rank:	\(0\)
2-Selmer rank:	\(2\)
Regulator:	\( 1 \)
Real period:	\( 6.632969 \)
Tamagawa product:	\( 1 \)
Torsion order:	\( 4 \)
Leading coefficient:	\( 3.731045 \)
Analytic order of Ш:	\( 9 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(8\)	\(8\)	\(1\)	\(1\)
\(37\)	\(2\)	\(2\)	\(1\)	\(1 + 10 T + 37 T^{2}\)

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?
\(2\)	2.240.1	yes
\(3\)	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
  \(g_4 = -\frac{1114}{11} b^{5} + \frac{25040}{11} b^{4} - \frac{61417}{11} b^{3} - \frac{44530}{11} b^{2} + \frac{117344}{11} b + \frac{58177}{11}\)
  \(g_6 = 3626 b^{5} + 49025 b^{4} - \frac{284049}{2} b^{3} - 210308 b^{2} + 329892 b + 363155\)
   Conductor norm: 16777216

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