Properties

Label 1312.c
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

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Genus 2 curves in isogeny class 1312.c

Label Equation
1312.c.671744.1 \(y^2 + (x + 1)y = x^6 + 4x^5 + 7x^4 + 5x^3 + 2x^2\)

L-function data

Analytic rank:\(0\)
 
Bad L-factors:
Prime L-Factor
\(2\)\( 1 - T\)
\(41\)\( ( 1 - T )( 1 + 2 T + 41 T^{2} )\)
 
Good L-factors:
Prime L-Factor
\(3\)\( 1 + 2 T + 4 T^{2} + 6 T^{3} + 9 T^{4}\)
\(5\)\( 1 + 2 T + 6 T^{2} + 10 T^{3} + 25 T^{4}\)
\(7\)\( 1 - 6 T^{2} + 49 T^{4}\)
\(11\)\( 1 + 2 T + 8 T^{2} + 22 T^{3} + 121 T^{4}\)
\(13\)\( 1 + 6 T^{2} + 169 T^{4}\)
\(17\)\( ( 1 - 6 T + 17 T^{2} )( 1 + 4 T + 17 T^{2} )\)
\(19\)\( 1 - 2 T + 8 T^{2} - 38 T^{3} + 361 T^{4}\)
\(23\)\( ( 1 - 2 T + 23 T^{2} )^{2}\)
\(29\)\( ( 1 - 8 T + 29 T^{2} )( 1 + 10 T + 29 T^{2} )\)
$\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.