Genus 2 isogeny class 277.a

• 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

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:

\(0\)

Bad L-factors:

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

Good L-factors:

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

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.