Properties

Label 62411.b
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 62411.b

Label Equation
62411.b.62411.1 \(y^2 + y = x^5 - x\)

L-function data

Analytic rank:\(3\)
 
Bad L-factors:
Prime L-Factor
\(139\)\( ( 1 + T )( 1 + 19 T + 139 T^{2} )\)
\(449\)\( ( 1 - T )( 1 - 5 T + 449 T^{2} )\)
 
Good L-factors:
Prime L-Factor
\(2\)\( ( 1 + 2 T^{2} )( 1 + 2 T + 2 T^{2} )\)
\(3\)\( 1 + 3 T + 7 T^{2} + 9 T^{3} + 9 T^{4}\)
\(5\)\( 1 + 5 T + 15 T^{2} + 25 T^{3} + 25 T^{4}\)
\(7\)\( ( 1 + 7 T^{2} )( 1 + 5 T + 7 T^{2} )\)
\(11\)\( 1 + T - 6 T^{2} + 11 T^{3} + 121 T^{4}\)
\(13\)\( ( 1 + 2 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} )\)
\(17\)\( ( 1 - 3 T + 17 T^{2} )( 1 + 6 T + 17 T^{2} )\)
\(19\)\( 1 + 11 T + 63 T^{2} + 209 T^{3} + 361 T^{4}\)
\(23\)\( 1 - 6 T + 42 T^{2} - 138 T^{3} + 529 T^{4}\)
\(29\)\( 1 - 8 T^{2} + 841 T^{4}\)
$\cdots$$\cdots$
 
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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.