Properties

Label 7403.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 7403.a

Label Equation
7403.a.7403.1 \(y^2 + y = x^5 + x^4 - 2x^3 - x^2 + x\)

L-function data

Analytic rank:\(2\)
 
Bad L-factors:
Prime L-Factor
\(11\)\( ( 1 + T )( 1 + 2 T + 11 T^{2} )\)
\(673\)\( ( 1 + T )( 1 + 34 T + 673 T^{2} )\)
 
Good L-factors:
Prime L-Factor
\(2\)\( ( 1 + 2 T^{2} )( 1 + 2 T + 2 T^{2} )\)
\(3\)\( 1 + 3 T + 7 T^{2} + 9 T^{3} + 9 T^{4}\)
\(5\)\( ( 1 - T + 5 T^{2} )( 1 + 4 T + 5 T^{2} )\)
\(7\)\( 1 + 4 T + 16 T^{2} + 28 T^{3} + 49 T^{4}\)
\(13\)\( 1 + 4 T + 6 T^{2} + 52 T^{3} + 169 T^{4}\)
\(17\)\( 1 - T + 7 T^{2} - 17 T^{3} + 289 T^{4}\)
\(19\)\( 1 + T - 13 T^{2} + 19 T^{3} + 361 T^{4}\)
\(23\)\( 1 + 5 T + 14 T^{2} + 115 T^{3} + 529 T^{4}\)
\(29\)\( 1 - 10 T^{2} + 841 T^{4}\)
$\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.