Genus 2 isogeny class 38416.a

• Label: 38416.a
• Conductor: $38416$
• Sato-Tate group: $E_1$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\mathrm{M}_2(\R)\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathrm{M}_2(\Q)\)
• \(\End(J) \otimes \Q\): \(\mathrm{M}_2(\Q)\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: no

L-function data

Analytic rank:

\(2\)  (upper bound)

Mordell-Weil rank:

\(2\)

Bad L-factors:

Prime	L-Factor
\(2\)	\( 1\)
\(7\)	\( 1\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(3\)	\( ( 1 + T + 3 T^{2} )^{2}\)	2.3.c_h
\(5\)	\( ( 1 + 3 T + 5 T^{2} )^{2}\)	2.5.g_t
\(11\)	\( ( 1 + 3 T + 11 T^{2} )^{2}\)	2.11.g_bf
\(13\)	\( ( 1 + 2 T + 13 T^{2} )^{2}\)	2.13.e_be
\(17\)	\( ( 1 + 3 T + 17 T^{2} )^{2}\)	2.17.g_br
\(19\)	\( ( 1 - T + 19 T^{2} )^{2}\)	2.19.ac_bn
\(23\)	\( ( 1 - 3 T + 23 T^{2} )^{2}\)	2.23.ag_cd
\(29\)	\( ( 1 + 6 T + 29 T^{2} )^{2}\)	2.29.m_dq
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Sato-Tate group

\(\mathrm{ST} =\) $E_1$, \(\quad \mathrm{ST}^0 = \mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over \(\Q\)

Decomposes up to isogeny as the square of the elliptic curve isogeny class:
  Elliptic curve isogeny class 196.a

Endomorphisms of the Jacobian

Not of \(\GL_2\)-type over \(\Q\)

Endomorphism algebra over \(\Q\):

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

All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).

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

Genus 2 curves in isogeny class 38416.a

Label	Equation
38416.a.614656.1	\(y^2 = x^6 - 3x^5 - x^4 + 7x^3 - x^2 - 3x + 1\)