Properties

Label 523.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 523.a

Label Equation
523.a.523.1 \(y^2 + (x + 1)y = x^5 - x^4 - x^3\)
523.a.523.2 \(y^2 + xy = x^5 - 31x^4 - 110x^3 + 21x^2 - x\)

L-function data

Analytic rank:\(0\)
 
Bad L-factors:
Prime L-Factor
\(523\)\( ( 1 + T )( 1 - 24 T + 523 T^{2} )\)
 
Good L-factors:
Prime L-Factor
\(2\)\( ( 1 - T + 2 T^{2} )( 1 + 2 T + 2 T^{2} )\)
\(3\)\( 1 + 9 T^{4}\)
\(5\)\( ( 1 + 5 T^{2} )( 1 + 4 T + 5 T^{2} )\)
\(7\)\( ( 1 - 4 T + 7 T^{2} )( 1 + 2 T + 7 T^{2} )\)
\(11\)\( 1 - 2 T + 2 T^{2} - 22 T^{3} + 121 T^{4}\)
\(13\)\( ( 1 - 2 T + 13 T^{2} )( 1 + T + 13 T^{2} )\)
\(17\)\( ( 1 - 5 T + 17 T^{2} )( 1 + 2 T + 17 T^{2} )\)
\(19\)\( 1 + T - 2 T^{2} + 19 T^{3} + 361 T^{4}\)
\(23\)\( ( 1 + T + 23 T^{2} )( 1 + 8 T + 23 T^{2} )\)
\(29\)\( ( 1 - 10 T + 29 T^{2} )( 1 + 9 T + 29 T^{2} )\)
$\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.