Properties

Label 277.a
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

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This isogeny class has the smallest prime conductor of any isogeny class of abelian surface, as proved by Brumer and Kramer in [10.1090/S0002-9947-2013-05909-0] .

Genus 2 curves in isogeny class 277.a

Label Equation
277.a.277.1 \(y^2 + (x^3 + x^2 + x + 1)y = -x^2 - x\)
277.a.277.2 \(y^2 + y = x^5 - 9x^4 + 14x^3 - 19x^2 + 11x - 6\)

L-function data

Analytic rank:\(0\)
 
Bad L-factors:
Prime L-Factor
\(277\)\( (1+T)(1-8T+277T^{2})\)
 
Good L-factors:
Prime L-Factor
\(2\)\( (1+2T^{2})(1+2T+2T^{2})\)
\(3\)\( 1+T+T^{2}+3T^{3}+9T^{4}\)
\(5\)\( (1-3T+5T^{2})(1+4T+5T^{2})\)
\(7\)\( 1-T+3T^{2}-7T^{3}+49T^{4}\)
\(11\)\( 1+2T+4T^{2}+22T^{3}+121T^{4}\)
\(13\)\( 1-3T+7T^{2}-39T^{3}+169T^{4}\)
\(17\)\( 1+4T+28T^{2}+68T^{3}+289T^{4}\)
\(19\)\( 1+T-22T^{2}+19T^{3}+361T^{4}\)
\(23\)\( 1-3T+22T^{2}-69T^{3}+529T^{4}\)
\(29\)\( 1+T+13T^{2}+29T^{3}+841T^{4}\)
$\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 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.