Properties

Label 1136.a
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

Related objects

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Genus 2 curves in isogeny class 1136.a

Label Equation
1136.a.9088.1 \(y^2 + (x^3 + x)y = x^4 - x^3 + 2x^2 - x + 1\)
1136.a.290816.1 \(y^2 + (x^3 + x^2)y = -5x^4 - 9x^3 + 25x^2 + 40x - 24\)

L-function data

Analytic rank:\(0\)
 
Bad L-factors:
Prime L-Factor
\(2\)\( 1 - T\)
\(71\)\( ( 1 + T )( 1 - 16 T + 71 T^{2} )\)
 
Good L-factors:
Prime L-Factor
\(3\)\( ( 1 + 3 T^{2} )( 1 + 3 T + 3 T^{2} )\)
\(5\)\( 1 + 2 T^{2} + 25 T^{4}\)
\(7\)\( 1 + T - 2 T^{2} + 7 T^{3} + 49 T^{4}\)
\(11\)\( 1 + 2 T - 6 T^{2} + 22 T^{3} + 121 T^{4}\)
\(13\)\( 1 + 3 T + 12 T^{2} + 39 T^{3} + 169 T^{4}\)
\(17\)\( 1 - 10 T^{2} + 289 T^{4}\)
\(19\)\( 1 - 3 T + 6 T^{2} - 57 T^{3} + 361 T^{4}\)
\(23\)\( ( 1 - 8 T + 23 T^{2} )( 1 - 3 T + 23 T^{2} )\)
\(29\)\( 1 - 4 T + 34 T^{2} - 116 T^{3} + 841 T^{4}\)
$\cdots$$\cdots$
 
See L-function page for more information

Sato-Tate group

\(\mathrm{ST} =\) $\mathrm{USp}(4)$

Endomorphisms of the Jacobian

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

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.