Properties

Label 5819.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 5819.a

Label Equation
5819.a.5819.1 \(y^2 + x^3y = x^4 - x^3 + 2x^2 - x + 1\)

L-function data

Analytic rank:\(2\)
 
Bad L-factors:
Prime L-Factor
\(11\)\( ( 1 - T )( 1 + 11 T^{2} )\)
\(23\)\( 1 + 4 T + 23 T^{2}\)
 
Good L-factors:
Prime L-Factor
\(2\)\( ( 1 + 2 T^{2} )( 1 + 2 T + 2 T^{2} )\)
\(3\)\( 1 + 3 T + 5 T^{2} + 9 T^{3} + 9 T^{4}\)
\(5\)\( 1 + 5 T + 13 T^{2} + 25 T^{3} + 25 T^{4}\)
\(7\)\( 1 + 4 T + 15 T^{2} + 28 T^{3} + 49 T^{4}\)
\(13\)\( 1 + 6 T + 25 T^{2} + 78 T^{3} + 169 T^{4}\)
\(17\)\( 1 + 2 T + 4 T^{2} + 34 T^{3} + 289 T^{4}\)
\(19\)\( 1 - T + 14 T^{2} - 19 T^{3} + 361 T^{4}\)
\(29\)\( 1 + T^{2} + 841 T^{4}\)
$\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.