Properties

Label 5170.a
Conductor $5170$
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\End(J) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

Related objects

Learn more

Genus 2 curves in isogeny class 5170.a

Label Equation
5170.a.10340.1 \(y^2 + (x^3 + 1)y = -3x^6 + 25x^4 + 14x^3 - 46x^2 - 22x + 24\)

L-function data

Analytic rank:\(0\)
Mordell-Weil rank:\(0\)
 
Bad L-factors:
Prime L-Factor
\(2\)\( ( 1 + T )( 1 - T + 2 T^{2} )\)
\(5\)\( ( 1 - T )( 1 + 5 T^{2} )\)
\(11\)\( ( 1 - T )( 1 + 11 T^{2} )\)
\(47\)\( ( 1 + T )( 1 + 47 T^{2} )\)
 
Good L-factors:
Prime L-Factor
\(3\)\( ( 1 - 2 T + 3 T^{2} )( 1 + 2 T + 3 T^{2} )\)
\(7\)\( ( 1 - 4 T + 7 T^{2} )( 1 + T + 7 T^{2} )\)
\(13\)\( ( 1 - 5 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} )\)
\(17\)\( ( 1 - 6 T + 17 T^{2} )( 1 + 6 T + 17 T^{2} )\)
\(19\)\( 1 - 2 T + 2 T^{2} - 38 T^{3} + 361 T^{4}\)
\(23\)\( 1 + 9 T + 58 T^{2} + 207 T^{3} + 529 T^{4}\)
\(29\)\( ( 1 - 6 T + 29 T^{2} )( 1 + 6 T + 29 T^{2} )\)
$\cdots$$\cdots$
 
See L-function page for more information

Sato-Tate group

\(\mathrm{ST} =\) $\mathrm{USp}(4)$

Decomposition of the Jacobian

Simple over \(\overline{\Q}\)

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.