Properties

Label 1309.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 1309.a

Label Equation
1309.a.9163.1 \(y^2 + (x^2 + 1)y = 7x^5 - x^4 - 5x^3 - x^2 + x\)
1309.a.9163.2 \(y^2 + x^2y = -x^6 + 20x^5 - 128x^4 + 248x^3 + 32x^2 + x\)

L-function data

Analytic rank:\(0\)
Mordell-Weil rank:\(0\)
 
Bad L-factors:
Prime L-Factor
\(7\)\( ( 1 - T )( 1 + 7 T^{2} )\)
\(11\)\( ( 1 + T )( 1 - 4 T + 11 T^{2} )\)
\(17\)\( ( 1 - T )( 1 + 6 T + 17 T^{2} )\)
 
Good L-factors:
Prime L-Factor
\(2\)\( 1 + T + 2 T^{3} + 4 T^{4}\)
\(3\)\( 1 - T + 2 T^{2} - 3 T^{3} + 9 T^{4}\)
\(5\)\( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4}\)
\(13\)\( 1 + 2 T - 6 T^{2} + 26 T^{3} + 169 T^{4}\)
\(19\)\( ( 1 - 4 T + 19 T^{2} )( 1 + 2 T + 19 T^{2} )\)
\(23\)\( ( 1 - 7 T + 23 T^{2} )( 1 + 23 T^{2} )\)
\(29\)\( ( 1 - 6 T + 29 T^{2} )( 1 + 10 T + 29 T^{2} )\)
$\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 \(\Q\)

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.