Properties

Label 394.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 394.a

Label Equation
394.a.394.1 \(y^2 + (x^3 + x)y = 2x^5 + x^4 - 12x^3 + 17x - 9\)
394.a.3152.1 \(y^2 + (x + 1)y = -x^5\)

L-function data

Analytic rank:\(0\)
 
Bad L-factors:
Prime L-Factor
\(2\)\( ( 1 - T )( 1 + 2 T + 2 T^{2} )\)
\(197\)\( ( 1 - T )( 1 - 18 T + 197 T^{2} )\)
 
Good L-factors:
Prime L-Factor
\(3\)\( 1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4}\)
\(5\)\( ( 1 + 5 T^{2} )( 1 + 4 T + 5 T^{2} )\)
\(7\)\( 1 - T - 2 T^{2} - 7 T^{3} + 49 T^{4}\)
\(11\)\( 1 - 2 T + 2 T^{2} - 22 T^{3} + 121 T^{4}\)
\(13\)\( 1 + 2 T + 22 T^{2} + 26 T^{3} + 169 T^{4}\)
\(17\)\( 1 + 2 T - 6 T^{2} + 34 T^{3} + 289 T^{4}\)
\(19\)\( 1 - T - 22 T^{2} - 19 T^{3} + 361 T^{4}\)
\(23\)\( 1 - 5 T + 30 T^{2} - 115 T^{3} + 529 T^{4}\)
\(29\)\( 1 + 3 T + 28 T^{2} + 87 T^{3} + 841 T^{4}\)
$\cdots$$\cdots$
 
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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.