# 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

# Related objects

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:$0$

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$

$\mathrm{ST} =$ $\mathrm{USp}(4)$
not of $\GL_2$-type over $\Q$
All endomorphisms of the Jacobian are defined over $\Q$