# Properties

 Label 277.a Conductor $277$ 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

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:$$0$$
Mordell-Weil rank:$$0$$

Prime L-Factor
$$277$$$$( 1 + T )( 1 - 8 T + 277 T^{2} )$$

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

## 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.