Properties

Label 4428.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 4428.a

Label Equation
4428.a.239112.1 \(y^2 + (x^3 + x + 1)y = -2x^5 - x^4 - x\)

L-function data

Analytic rank:\(1\)
Mordell-Weil rank:\(1\)
 
Bad L-factors:
Prime L-Factor
\(2\)\( 1 + T + T^{2}\)
\(3\)\( 1 + T\)
\(41\)\( ( 1 + T )( 1 + 3 T + 41 T^{2} )\)
 
Good L-factors:
Prime L-Factor
\(5\)\( 1 + 2 T + T^{2} + 10 T^{3} + 25 T^{4}\)
\(7\)\( ( 1 + T + 7 T^{2} )( 1 + 4 T + 7 T^{2} )\)
\(11\)\( ( 1 + 11 T^{2} )( 1 + 4 T + 11 T^{2} )\)
\(13\)\( 1 + 2 T + 26 T^{3} + 169 T^{4}\)
\(17\)\( 1 - T + 10 T^{2} - 17 T^{3} + 289 T^{4}\)
\(19\)\( 1 - 20 T^{2} + 361 T^{4}\)
\(23\)\( 1 - T - 14 T^{2} - 23 T^{3} + 529 T^{4}\)
\(29\)\( 1 + 2 T + 22 T^{2} + 58 T^{3} + 841 T^{4}\)
$\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 \(\Q\)

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.