Properties

Label 100010.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

Learn more about

Genus 2 curves in isogeny class 100010.a

Label Equation
100010.a.400040.1 \(y^2 + (x^3 + 1)y = -x^6 + x^5 + 2x^4 + x^3 - 10x^2 + 2x + 5\)

L-function data

Analytic rank:\(1\)
 
Bad L-factors:
Prime L-Factor
\(2\)\( ( 1 - T )( 1 - T + 2 T^{2} )\)
\(5\)\( ( 1 - T )( 1 + 2 T + 5 T^{2} )\)
\(73\)\( ( 1 - T )( 1 + 10 T + 73 T^{2} )\)
\(137\)\( ( 1 + T )( 1 - 10 T + 137 T^{2} )\)
 
Good L-factors:
Prime L-Factor
\(3\)\( 1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4}\)
\(7\)\( 1 + 3 T + 6 T^{2} + 21 T^{3} + 49 T^{4}\)
\(11\)\( 1 - 10 T^{2} + 121 T^{4}\)
\(13\)\( 1 - 6 T + 22 T^{2} - 78 T^{3} + 169 T^{4}\)
\(17\)\( 1 - 4 T + 18 T^{2} - 68 T^{3} + 289 T^{4}\)
\(19\)\( ( 1 - 4 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} )\)
\(23\)\( 1 - 7 T + 54 T^{2} - 161 T^{3} + 529 T^{4}\)
\(29\)\( 1 - 10 T^{2} + 841 T^{4}\)
$\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.