Properties

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

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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.