Properties

Label 464.a
Conductor $464$
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 464.a

Label Equation
464.a.464.1 \(y^2 + (x + 1)y = -x^6 - 2x^5 - 2x^4 - x^3\)
464.a.29696.1 \(y^2 + (x + 1)y = 8x^5 + 3x^4 - 4x^3 - 2x^2\)
464.a.29696.2 \(y^2 + xy = 4x^5 + 33x^4 + 72x^3 + 16x^2 + x\)

L-function data

Analytic rank:\(0\)
Mordell-Weil rank:\(0\)
 
Bad L-factors:
Prime L-Factor
\(2\)\( 1 + T + 2 T^{2}\)
\(29\)\( ( 1 + T )( 1 + 2 T + 29 T^{2} )\)
 
Good L-factors:
Prime L-Factor
\(3\)\( 1 + T + 2 T^{2} + 3 T^{3} + 9 T^{4}\)
\(5\)\( 1 + T + 5 T^{3} + 25 T^{4}\)
\(7\)\( 1 + 2 T - 2 T^{2} + 14 T^{3} + 49 T^{4}\)
\(11\)\( 1 - 5 T + 18 T^{2} - 55 T^{3} + 121 T^{4}\)
\(13\)\( 1 + 3 T + 12 T^{2} + 39 T^{3} + 169 T^{4}\)
\(17\)\( 1 - 2 T + 2 T^{2} - 34 T^{3} + 289 T^{4}\)
\(19\)\( ( 1 - 4 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )\)
\(23\)\( ( 1 - 8 T + 23 T^{2} )( 1 + 4 T + 23 T^{2} )\)
$\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.