Properties

Label 2457.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

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Genus 2 curves in isogeny class 2457.a

Label Equation
2457.a.95823.1 \(y^2 + (x^3 + 1)y = x^5 - x^4 - 3x^3 + 6x^2 - 6x + 2\)
2457.a.154791.1 \(y^2 + (x^3 + 1)y = x^5 - 3x^3 - x^2 + 2\)

L-function data

Analytic rank:\(1\)
Mordell-Weil rank:\(1\)
 
Bad L-factors:
Prime L-Factor
\(3\)\( 1 + T\)
\(7\)\( ( 1 - T )( 1 + 4 T + 7 T^{2} )\)
\(13\)\( ( 1 + T )( 1 + 6 T + 13 T^{2} )\)
 
Good L-factors:
Prime L-Factor
\(2\)\( 1 + 2 T + 3 T^{2} + 4 T^{3} + 4 T^{4}\)
\(5\)\( ( 1 - 2 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} )\)
\(11\)\( 1 + 2 T^{2} + 121 T^{4}\)
\(17\)\( 1 - 10 T^{2} + 289 T^{4}\)
\(19\)\( ( 1 - 4 T + 19 T^{2} )( 1 + 4 T + 19 T^{2} )\)
\(23\)\( 1 + 2 T + 6 T^{2} + 46 T^{3} + 529 T^{4}\)
\(29\)\( ( 1 - 10 T + 29 T^{2} )( 1 + 8 T + 29 T^{2} )\)
$\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.