Properties

Label 708.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 708.a

Label Equation
708.a.2832.1 \(y^2 + (x^2 + x + 1)y = x^5\)
708.a.19116.1 \(y^2 + (x^3 + 1)y = -x^5 + 4x^2 + 4x - 1\)
708.a.181248.1 \(y^2 + (x^3 + 1)y = -x^6 - 4x^5 + 9x^4 + 48x^3 - 41x^2 - 98x - 36\)

L-function data

Analytic rank:\(0\)
 
Bad L-factors:
Prime L-Factor
\(2\)\( 1 + T^{2}\)
\(3\)\( ( 1 + T )( 1 + T + 3 T^{2} )\)
\(59\)\( ( 1 + T )( 1 + 59 T^{2} )\)
 
Good L-factors:
Prime L-Factor
\(5\)\( 1 - T - 5 T^{3} + 25 T^{4}\)
\(7\)\( 1 + 3 T + 6 T^{2} + 21 T^{3} + 49 T^{4}\)
\(11\)\( 1 - 2 T + 2 T^{2} - 22 T^{3} + 121 T^{4}\)
\(13\)\( ( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} )\)
\(17\)\( ( 1 - 6 T + 17 T^{2} )( 1 + 2 T + 17 T^{2} )\)
\(19\)\( 1 + 5 T + 18 T^{2} + 95 T^{3} + 361 T^{4}\)
\(23\)\( 1 - 8 T + 42 T^{2} - 184 T^{3} + 529 T^{4}\)
\(29\)\( 1 + 5 T + 28 T^{2} + 145 T^{3} + 841 T^{4}\)
$\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.