Properties

Label 36864.f
Conductor $36864$
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 36864.f

Label Equation
36864.f.442368.1 \(y^2 = x^5 - 3x^4 - x^3 + 9x^2 - 6x\)

L-function data

Analytic rank:\(1\)
Mordell-Weil rank:\(1\)
 
Bad L-factors:
Prime L-Factor
\(2\)\( 1\)
\(3\)\( 1 + 3 T^{2}\)
 
Good L-factors:
Prime L-Factor
\(5\)\( 1 - 2 T^{2} + 25 T^{4}\)
\(7\)\( 1 + 2 T - 2 T^{2} + 14 T^{3} + 49 T^{4}\)
\(11\)\( 1 + 4 T + 6 T^{2} + 44 T^{3} + 121 T^{4}\)
\(13\)\( 1 + 6 T^{2} + 169 T^{4}\)
\(17\)\( ( 1 - 2 T + 17 T^{2} )( 1 + 4 T + 17 T^{2} )\)
\(19\)\( ( 1 - 8 T + 19 T^{2} )( 1 + 4 T + 19 T^{2} )\)
\(23\)\( ( 1 + 23 T^{2} )( 1 + 8 T + 23 T^{2} )\)
\(29\)\( 1 - 4 T + 38 T^{2} - 116 T^{3} + 841 T^{4}\)
$\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.