Properties

Label 762.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 762.a

Label Equation
762.a.3048.1 \(y^2 + (x^3 + x^2 + x)y = x^2 + x + 1\)
762.a.82296.1 \(y^2 + (x^2 + x)y = x^5 - 8x^4 + 14x^3 + 2x^2 - x\)

L-function data

Analytic rank:\(0\)
 
Bad L-factors:
Prime L-Factor
\(2\)\( ( 1 - T )( 1 + T + 2 T^{2} )\)
\(3\)\( ( 1 - T )( 1 + 2 T + 3 T^{2} )\)
\(127\)\( ( 1 + T )( 1 + 16 T + 127 T^{2} )\)
 
Good L-factors:
Prime L-Factor
\(5\)\( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4}\)
\(7\)\( 1 - T + 6 T^{2} - 7 T^{3} + 49 T^{4}\)
\(11\)\( 1 + 3 T + 10 T^{2} + 33 T^{3} + 121 T^{4}\)
\(13\)\( 1 + 2 T - 6 T^{2} + 26 T^{3} + 169 T^{4}\)
\(17\)\( 1 - T + 16 T^{2} - 17 T^{3} + 289 T^{4}\)
\(19\)\( 1 - 5 T + 26 T^{2} - 95 T^{3} + 361 T^{4}\)
\(23\)\( 1 - T - 2 T^{2} - 23 T^{3} + 529 T^{4}\)
\(29\)\( ( 1 - 6 T + 29 T^{2} )( 1 + 6 T + 29 T^{2} )\)
$\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.