Properties

Label 100224.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 100224.a

Label Equation
100224.a.601344.1 \(y^2 = x^5 + 2x^4 + 4x^3 + 2x^2 + x - 1\)

L-function data

Analytic rank:\(1\)
 
Bad L-factors:
Prime L-Factor
\(2\)\( 1\)
\(3\)\( 1 - T\)
\(29\)\( ( 1 + T )( 1 + 4 T + 29 T^{2} )\)
 
Good L-factors:
Prime L-Factor
\(5\)\( 1 + 2 T^{2} + 25 T^{4}\)
\(7\)\( ( 1 - T + 7 T^{2} )( 1 + 4 T + 7 T^{2} )\)
\(11\)\( 1 + T + 6 T^{2} + 11 T^{3} + 121 T^{4}\)
\(13\)\( 1 + 5 T + 28 T^{2} + 65 T^{3} + 169 T^{4}\)
\(17\)\( 1 + T - 2 T^{2} + 17 T^{3} + 289 T^{4}\)
\(19\)\( ( 1 + 19 T^{2} )^{2}\)
\(23\)\( ( 1 - 6 T + 23 T^{2} )( 1 + 8 T + 23 T^{2} )\)
$\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.