Properties

Label 1051.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 1051.a

Label Equation
1051.a.1051.1 \(y^2 + y = x^5 - x^4 + x^2 - x\)

L-function data

Analytic rank:\(1\)
 
Bad L-factors:
Prime L-Factor
\(1051\)\( ( 1 - T )( 1 + 36 T + 1051 T^{2} )\)
 
Good L-factors:
Prime L-Factor
\(2\)\( ( 1 + 2 T^{2} )( 1 + 2 T + 2 T^{2} )\)
\(3\)\( ( 1 + 3 T^{2} )( 1 + 3 T + 3 T^{2} )\)
\(5\)\( 1 + T - T^{2} + 5 T^{3} + 25 T^{4}\)
\(7\)\( ( 1 - 2 T + 7 T^{2} )( 1 + 5 T + 7 T^{2} )\)
\(11\)\( 1 + T - T^{2} + 11 T^{3} + 121 T^{4}\)
\(13\)\( 1 - T + 8 T^{2} - 13 T^{3} + 169 T^{4}\)
\(17\)\( ( 1 - 6 T + 17 T^{2} )( 1 + 17 T^{2} )\)
\(19\)\( ( 1 + 4 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )\)
\(23\)\( 1 - T + 18 T^{2} - 23 T^{3} + 529 T^{4}\)
\(29\)\( 1 + 7 T + 32 T^{2} + 203 T^{3} + 841 T^{4}\)
$\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.