Properties

Label 10028.a
Conductor $10028$
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\End(J) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

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Genus 2 curves in isogeny class 10028.a

Label Equation
10028.a.641792.1 \(y^2 + (x^3 + x^2 + x)y = 2x^4 + 2x^3 + 6x^2 + 2x + 4\)

L-function data

Analytic rank:\(1\)
Mordell-Weil rank:\(1\)
 
Bad L-factors:
Prime L-Factor
\(2\)\( ( 1 - T )( 1 + T )\)
\(23\)\( ( 1 + T )( 1 + 2 T + 23 T^{2} )\)
\(109\)\( ( 1 + T )( 1 - 7 T + 109 T^{2} )\)
 
Good L-factors:
Prime L-Factor
\(3\)\( 1 + 3 T + 7 T^{2} + 9 T^{3} + 9 T^{4}\)
\(5\)\( ( 1 - T + 5 T^{2} )( 1 + 3 T + 5 T^{2} )\)
\(7\)\( ( 1 - 3 T + 7 T^{2} )( 1 + 2 T + 7 T^{2} )\)
\(11\)\( 1 + 3 T + T^{2} + 33 T^{3} + 121 T^{4}\)
\(13\)\( 1 + 19 T^{2} + 169 T^{4}\)
\(17\)\( 1 + 2 T + 7 T^{2} + 34 T^{3} + 289 T^{4}\)
\(19\)\( 1 + 5 T + 33 T^{2} + 95 T^{3} + 361 T^{4}\)
\(29\)\( 1 + 3 T + 5 T^{2} + 87 T^{3} + 841 T^{4}\)
$\cdots$$\cdots$
 
See L-function page for more information

Sato-Tate group

\(\mathrm{ST} =\) $\mathrm{USp}(4)$

Decomposition of the Jacobian

Simple over \(\overline{\Q}\)

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.