Properties

Label 5200.b
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 5200.b

Label Equation
5200.b.332800.1 \(y^2 + (x + 1)y = x^6 + 2x^5 + 2x^4 - x^2\)

L-function data

Analytic rank:\(0\)
Mordell-Weil rank:\(0\)
 
Bad L-factors:
Prime L-Factor
\(2\)\( 1 - T\)
\(5\)\( 1 + T^{2}\)
\(13\)\( ( 1 - T )( 1 - T + 13 T^{2} )\)
 
Good L-factors:
Prime L-Factor
\(3\)\( 1 + T - T^{2} + 3 T^{3} + 9 T^{4}\)
\(7\)\( 1 + 2 T^{2} + 49 T^{4}\)
\(11\)\( 1 + 4 T + 12 T^{2} + 44 T^{3} + 121 T^{4}\)
\(17\)\( 1 - T + 27 T^{2} - 17 T^{3} + 289 T^{4}\)
\(19\)\( ( 1 - 3 T + 19 T^{2} )( 1 + 6 T + 19 T^{2} )\)
\(23\)\( 1 - 3 T + 10 T^{2} - 69 T^{3} + 529 T^{4}\)
\(29\)\( 1 + 11 T + 63 T^{2} + 319 T^{3} + 841 T^{4}\)
$\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 \(\Q\)

Endomorphism algebra over \(\Q\):

\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R\)

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.