Properties

Label 6544.b
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 6544.b

Label Equation
6544.b.418816.1 \(y^2 + (x^3 + x)y = x^5 + 2x^2 + x + 1\)

L-function data

Analytic rank:\(0\)
Mordell-Weil rank:\(0\)
 
Bad L-factors:
Prime L-Factor
\(2\)\( 1 - T\)
\(409\)\( ( 1 - T )( 1 + 19 T + 409 T^{2} )\)
 
Good L-factors:
Prime L-Factor
\(3\)\( 1 + T - T^{2} + 3 T^{3} + 9 T^{4}\)
\(5\)\( 1 + T + 7 T^{2} + 5 T^{3} + 25 T^{4}\)
\(7\)\( 1 + 2 T^{2} + 49 T^{4}\)
\(11\)\( ( 1 - 4 T + 11 T^{2} )( 1 + T + 11 T^{2} )\)
\(13\)\( 1 + 12 T^{2} + 169 T^{4}\)
\(17\)\( ( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} )\)
\(19\)\( 1 + 28 T^{2} + 361 T^{4}\)
\(23\)\( 1 + 5 T + 26 T^{2} + 115 T^{3} + 529 T^{4}\)
\(29\)\( 1 - 3 T + 28 T^{2} - 87 T^{3} + 841 T^{4}\)
$\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.