Properties

Label 2980.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 2980.a

Label Equation
2980.a.381440.1 \(y^2 + (x^3 + 1)y = 2x^5 + 3x^4 - x^2 - x\)

L-function data

Analytic rank:\(1\)
Mordell-Weil rank:\(1\)
 
Bad L-factors:
Prime L-Factor
\(2\)\( ( 1 - T )( 1 + T )\)
\(5\)\( ( 1 - T )( 1 + 4 T + 5 T^{2} )\)
\(149\)\( ( 1 - T )( 1 + 6 T + 149 T^{2} )\)
 
Good L-factors:
Prime L-Factor
\(3\)\( 1 + 4 T + 8 T^{2} + 12 T^{3} + 9 T^{4}\)
\(7\)\( 1 + 4 T + 16 T^{2} + 28 T^{3} + 49 T^{4}\)
\(11\)\( 1 - 8 T^{2} + 121 T^{4}\)
\(13\)\( 1 + 3 T + 20 T^{2} + 39 T^{3} + 169 T^{4}\)
\(17\)\( 1 - T + 12 T^{2} - 17 T^{3} + 289 T^{4}\)
\(19\)\( 1 + 3 T + 18 T^{2} + 57 T^{3} + 361 T^{4}\)
\(23\)\( 1 - 3 T - 4 T^{2} - 69 T^{3} + 529 T^{4}\)
\(29\)\( ( 1 - 6 T + 29 T^{2} )( 1 + 6 T + 29 T^{2} )\)
$\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.