Properties

Label 10075.b
Conductor $10075$
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 10075.b

Label Equation
10075.b.654875.1 \(y^2 + xy = x^5 + 2x^4 - 4x^3 - 8x^2 - 1\)

L-function data

Analytic rank:\(0\)
Mordell-Weil rank:\(0\)
 
Bad L-factors:
Prime L-Factor
\(5\)\( 1 + 2 T + 5 T^{2}\)
\(13\)\( ( 1 + T )( 1 - 6 T + 13 T^{2} )\)
\(31\)\( ( 1 - T )( 1 + 4 T + 31 T^{2} )\)
 
Good L-factors:
Prime L-Factor
\(2\)\( 1 + T + 2 T^{3} + 4 T^{4}\)
\(3\)\( 1 - T - 2 T^{2} - 3 T^{3} + 9 T^{4}\)
\(7\)\( ( 1 - 4 T + 7 T^{2} )( 1 + 2 T + 7 T^{2} )\)
\(11\)\( ( 1 - 4 T + 11 T^{2} )( 1 + 6 T + 11 T^{2} )\)
\(17\)\( 1 - T + 16 T^{2} - 17 T^{3} + 289 T^{4}\)
\(19\)\( ( 1 - 5 T + 19 T^{2} )( 1 + 19 T^{2} )\)
\(23\)\( ( 1 - 8 T + 23 T^{2} )( 1 + 4 T + 23 T^{2} )\)
\(29\)\( ( 1 + 2 T + 29 T^{2} )^{2}\)
$\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.