Properties

Label 2059.a
Conductor $2059$
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

Related objects

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Genus 2 curves in isogeny class 2059.a

Label Equation
2059.a.2059.1 \(y^2 + y = x^5 - 4x^4 + 4x^3 - x\)

L-function data

Analytic rank:\(1\)
Mordell-Weil rank:\(1\)
 
Bad L-factors:
Prime L-Factor
\(29\)\( ( 1 + T )( 1 - 9 T + 29 T^{2} )\)
\(71\)\( ( 1 - T )( 1 + 9 T + 71 T^{2} )\)
 
Good L-factors:
Prime L-Factor
\(2\)\( ( 1 + 2 T^{2} )( 1 + 2 T + 2 T^{2} )\)
\(3\)\( ( 1 - T + 3 T^{2} )( 1 + 2 T + 3 T^{2} )\)
\(5\)\( 1 + T + T^{2} + 5 T^{3} + 25 T^{4}\)
\(7\)\( 1 + 4 T + 8 T^{2} + 28 T^{3} + 49 T^{4}\)
\(11\)\( 1 + T + 13 T^{2} + 11 T^{3} + 121 T^{4}\)
\(13\)\( ( 1 - 6 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} )\)
\(17\)\( 1 + 3 T + 19 T^{2} + 51 T^{3} + 289 T^{4}\)
\(19\)\( 1 + 2 T + 12 T^{2} + 38 T^{3} + 361 T^{4}\)
\(23\)\( ( 1 - 6 T + 23 T^{2} )( 1 + 5 T + 23 T^{2} )\)
$\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.