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

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

\(\mathrm{ST} =\) $\mathrm{USp}(4)$

Endomorphisms of the Jacobian

not of \(\GL_2\)-type over \(\Q\)

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.