Properties

Label 1721.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 1721.a

Label Equation
1721.a.1721.1 \(y^2 + (x^3 + 1)y = x^2 - x\)

L-function data

Analytic rank:\(1\)
 
Bad L-factors:
Prime L-Factor
\(1721\)\( ( 1 - T )( 1 + 39 T + 1721 T^{2} )\)
 
Good L-factors:
Prime L-Factor
\(2\)\( 1 + 2 T + 3 T^{2} + 4 T^{3} + 4 T^{4}\)
\(3\)\( ( 1 + 3 T^{2} )( 1 + 2 T + 3 T^{2} )\)
\(5\)\( ( 1 - T + 5 T^{2} )( 1 + 3 T + 5 T^{2} )\)
\(7\)\( 1 + 4 T^{2} + 49 T^{4}\)
\(11\)\( 1 + 4 T + 23 T^{2} + 44 T^{3} + 121 T^{4}\)
\(13\)\( 1 - T + 8 T^{2} - 13 T^{3} + 169 T^{4}\)
\(17\)\( 1 - T + 20 T^{2} - 17 T^{3} + 289 T^{4}\)
\(19\)\( 1 - 2 T - 7 T^{2} - 38 T^{3} + 361 T^{4}\)
\(23\)\( 1 + 2 T + 12 T^{2} + 46 T^{3} + 529 T^{4}\)
\(29\)\( 1 + T - 42 T^{2} + 29 T^{3} + 841 T^{4}\)
$\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 \(\Q\)

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.