Genus 2 isogeny class 196.a

• Label: 196.a
• Conductor: $196$
• 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

Genus 2 curves in isogeny class 196.a

Label	Equation
196.a.21952.1	\(y^2 + (x^2 + x)y = x^6 + 3x^5 + 6x^4 + 7x^3 + 6x^2 + 3x + 1\)

L-function data

Analytic rank:

\(0\)

Mordell-Weil rank:

\(0\)

Bad L-factors:

Prime	L-Factor
\(2\)	\( ( 1 + T )^{2}\)
\(7\)	\( ( 1 - T )^{2}\)

Good L-factors:

Prime	L-Factor
\(3\)	\( ( 1 + 2 T + 3 T^{2} )^{2}\)
\(5\)	\( ( 1 + 5 T^{2} )^{2}\)
\(11\)	\( ( 1 + 11 T^{2} )^{2}\)
\(13\)	\( ( 1 + 4 T + 13 T^{2} )^{2}\)
\(17\)	\( ( 1 - 6 T + 17 T^{2} )^{2}\)
\(19\)	\( ( 1 - 2 T + 19 T^{2} )^{2}\)
\(23\)	\( ( 1 + 23 T^{2} )^{2}\)
\(29\)	\( ( 1 + 6 T + 29 T^{2} )^{2}\)
$\cdots$	$\cdots$

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 14.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.