Genus 2 isogeny class 391876.b

• Label: 391876.b
• Conductor: $391876$
• 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

Genus 2 curves in isogeny class 391876.b

Label	Equation
391876.b.783752.1	\(y^2 + (x^3 + x + 1)y = -x^6 - 2x^5 + 3x^3 - x^2 - 3x - 1\)

L-function data

Analytic rank:

\(0\)

Mordell-Weil rank:

\(0\)

Bad L-factors:

Prime	L-Factor
\(2\)	\( 1 - T + T^{2}\)
\(313\)	\( 1 + 13 T + 313 T^{2}\)

Good L-factors:

Prime	L-Factor
\(3\)	\( 1 + T - 2 T^{2} + 3 T^{3} + 9 T^{4}\)
\(5\)	\( 1 + 2 T^{2} + 25 T^{4}\)
\(7\)	\( ( 1 - 7 T^{2} )^{2}\)
\(11\)	\( 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4}\)
\(13\)	\( 1 + 4 T + 3 T^{2} + 52 T^{3} + 169 T^{4}\)
\(17\)	\( 1 - 3 T + 20 T^{2} - 51 T^{3} + 289 T^{4}\)
\(19\)	\( ( 1 - T + 19 T^{2} )^{2}\)
\(23\)	\( ( 1 - 23 T^{2} )^{2}\)
\(29\)	\( 1 - 6 T + 7 T^{2} - 174 T^{3} + 841 T^{4}\)
$\cdots$	$\cdots$

See L-function page for more information

Sato-Tate group

\(\mathrm{ST} =\) $E_6$, \(\quad \mathrm{ST}^0 = \mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) 6.6.3004150512793.1 with defining polynomial:
  \(x^{6} - x^{5} - 130 x^{4} + 481 x^{3} + 2170 x^{2} - 6196 x - 15304\)

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
  \(g_4 = \frac{565236}{269} b^{5} - \frac{6779127}{1076} b^{4} - \frac{280105729}{1076} b^{3} + \frac{824114745}{538} b^{2} + \frac{1585191241}{1076} b - \frac{8578763427}{538}\)
  \(g_6 = -\frac{631755361055}{2152} b^{5} + \frac{1903087063619}{2152} b^{4} + \frac{39149248059607}{1076} b^{3} - \frac{461440519641675}{2152} b^{2} - \frac{221159457020257}{1076} b + \frac{600559022448280}{269}\)
   Conductor norm: 64

Endomorphisms of the Jacobian

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

Endomorphism algebra over \(\Q\):

\(\End (J_{}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{-3}) \)
\(\End (J_{}) \otimes \R\)	\(\simeq\)	\(\C\)

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) 6.6.3004150512793.1 with defining polynomial \(x^{6} - x^{5} - 130 x^{4} + 481 x^{3} + 2170 x^{2} - 6196 x - 15304\)

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

\(\End (J_{\overline{\Q}}) \otimes \Q \)	\(\simeq\)	\(\mathrm{M}_2(\)\(\Q\)\()\)
\(\End (J_{\overline{\Q}}) \otimes \R\)	\(\simeq\)	\(\mathrm{M}_2 (\R)\)

More complete information on endomorphism algebras and rings can be found on the pages of the individual curves in the isogeny class.