Properties

Label 1403.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 1403.a

Label Equation
1403.a.1403.1 \(y^2 + y = x^5 + x^4 - x^3 - x^2\)

L-function data

Analytic rank:\(1\)
 
Bad L-factors:
Prime L-Factor
\(23\)\( ( 1 + T )( 1 + 2 T + 23 T^{2} )\)
\(61\)\( ( 1 - T )( 1 + 6 T + 61 T^{2} )\)
 
Good L-factors:
Prime L-Factor
\(2\)\( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4}\)
\(3\)\( 1 + 3 T + 5 T^{2} + 9 T^{3} + 9 T^{4}\)
\(5\)\( ( 1 - 2 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} )\)
\(7\)\( 1 + T + 7 T^{2} + 7 T^{3} + 49 T^{4}\)
\(11\)\( 1 + 3 T + 15 T^{2} + 33 T^{3} + 121 T^{4}\)
\(13\)\( 1 + 3 T + 21 T^{2} + 39 T^{3} + 169 T^{4}\)
\(17\)\( 1 - 3 T + 35 T^{2} - 51 T^{3} + 289 T^{4}\)
\(19\)\( 1 - 5 T + 22 T^{2} - 95 T^{3} + 361 T^{4}\)
\(29\)\( ( 1 - 8 T + 29 T^{2} )( 1 + 6 T + 29 T^{2} )\)
$\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.