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].

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	Isogeny Class over \(\mathbb{F}_p\)
\(2\)	\( ( 1 + 2 T^{2} )( 1 + 2 T + 2 T^{2} )\)	2.2.c_e
\(3\)	\( 1 + T + T^{2} + 3 T^{3} + 9 T^{4}\)	2.3.b_b
\(5\)	\( ( 1 - 3 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} )\)	2.5.b_ac
\(7\)	\( 1 - T + 3 T^{2} - 7 T^{3} + 49 T^{4}\)	2.7.ab_d
\(11\)	\( 1 + 2 T + 4 T^{2} + 22 T^{3} + 121 T^{4}\)	2.11.c_e
\(13\)	\( 1 - 3 T + 7 T^{2} - 39 T^{3} + 169 T^{4}\)	2.13.ad_h
\(17\)	\( 1 + 4 T + 28 T^{2} + 68 T^{3} + 289 T^{4}\)	2.17.e_bc
\(19\)	\( 1 + T - 22 T^{2} + 19 T^{3} + 361 T^{4}\)	2.19.b_aw
\(23\)	\( 1 - 3 T + 22 T^{2} - 69 T^{3} + 529 T^{4}\)	2.23.ad_w
\(29\)	\( 1 + T + 13 T^{2} + 29 T^{3} + 841 T^{4}\)	2.29.b_n
$\cdots$	$\cdots$	$\cdots$

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.

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	\(y^2 + y = x^5 - 9x^4 + 14x^3 - 19x^2 + 11x - 6\)