Properties

Label 704.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 704.a

Label Equation
704.a.45056.1 \(y^2 + y = 4x^5 + 4x^4 - x^3 - 2x^2\)

L-function data

Analytic rank:\(0\)
 
Bad L-factors:
Prime L-Factor
\(2\)\( 1 + 2 T^{2}\)
\(11\)\( ( 1 - T )( 1 + 11 T^{2} )\)
 
Good L-factors:
Prime L-Factor
\(3\)\( 1 + T - 2 T^{2} + 3 T^{3} + 9 T^{4}\)
\(5\)\( ( 1 - 2 T + 5 T^{2} )( 1 + 3 T + 5 T^{2} )\)
\(7\)\( ( 1 - 2 T + 7 T^{2} )( 1 + 4 T + 7 T^{2} )\)
\(13\)\( ( 1 - 2 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} )\)
\(17\)\( ( 1 - 6 T + 17 T^{2} )( 1 + 6 T + 17 T^{2} )\)
\(19\)\( 1 - 2 T^{2} + 361 T^{4}\)
\(23\)\( 1 - 3 T - 2 T^{2} - 69 T^{3} + 529 T^{4}\)
\(29\)\( ( 1 - 8 T + 29 T^{2} )( 1 + 6 T + 29 T^{2} )\)
$\cdots$$\cdots$
 
See L-function page for more information

Sato-Tate group

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

Endomorphisms of the Jacobian

Not of \(\GL_2\)-type over \(\Q\)

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.