Properties

Label 925.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 925.a

Label Equation
925.a.925.1 \(y^2 + (x + 1)y = -x^5 + 2x^4 - x^3 - x^2\)
925.a.23125.1 \(y^2 + xy = 5x^5 + x^4 - 19x^3 + 18x^2 - 5x\)

L-function data

Analytic rank:\(0\)
 
Bad L-factors:
Prime L-Factor
\(5\)\( ( 1 - T )( 1 + T )\)
\(37\)\( ( 1 - T )( 1 - 6 T + 37 T^{2} )\)
 
Good L-factors:
Prime L-Factor
\(2\)\( 1 + T + 2 T^{3} + 4 T^{4}\)
\(3\)\( 1 + T + 2 T^{2} + 3 T^{3} + 9 T^{4}\)
\(7\)\( 1 - T + 6 T^{2} - 7 T^{3} + 49 T^{4}\)
\(11\)\( 1 + T - 6 T^{2} + 11 T^{3} + 121 T^{4}\)
\(13\)\( 1 + 2 T - 6 T^{2} + 26 T^{3} + 169 T^{4}\)
\(17\)\( 1 - 2 T + 2 T^{2} - 34 T^{3} + 289 T^{4}\)
\(19\)\( ( 1 - 4 T + 19 T^{2} )( 1 + 4 T + 19 T^{2} )\)
\(23\)\( ( 1 - 4 T + 23 T^{2} )( 1 + 8 T + 23 T^{2} )\)
\(29\)\( 1 - 2 T - 14 T^{2} - 58 T^{3} + 841 T^{4}\)
$\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.