Properties

Label 249.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 249.a

Label Equation
249.a.249.1 \(y^2 + (x^3 + 1)y = x^2 + x\)
249.a.6723.1 \(y^2 + (x^3 + 1)y = -x^5 + x^3 + x^2 + 3x + 2\)

L-function data

Analytic rank:\(0\)
 
Bad L-factors:
Prime L-Factor
\(3\)\( ( 1 - T )( 1 + 3 T + 3 T^{2} )\)
\(83\)\( ( 1 - T )( 1 + 83 T^{2} )\)
 
Good L-factors:
Prime L-Factor
\(2\)\( 1 + 2 T + 3 T^{2} + 4 T^{3} + 4 T^{4}\)
\(5\)\( 1 + 2 T^{2} + 25 T^{4}\)
\(7\)\( 1 + T - 2 T^{2} + 7 T^{3} + 49 T^{4}\)
\(11\)\( ( 1 - 5 T + 11 T^{2} )( 1 + 4 T + 11 T^{2} )\)
\(13\)\( 1 - 2 T^{2} + 169 T^{4}\)
\(17\)\( ( 1 - 2 T + 17 T^{2} )( 1 + 3 T + 17 T^{2} )\)
\(19\)\( ( 1 - 2 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )\)
\(23\)\( 1 - 26 T^{2} + 529 T^{4}\)
\(29\)\( 1 - 3 T + 32 T^{2} - 87 T^{3} + 841 T^{4}\)
$\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.