Genus 2 isogeny class 28561.a

• Label: 28561.a
• Conductor: $28561$
• Sato-Tate group: $F_{ac}$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\C \times \C\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathsf{CM}\)
• \(\End(J) \otimes \Q\): \(\Q\)
• \(\overline{\Q}\)-simple: yes
• \(\mathrm{GL}_2\)-type: no

Genus 2 curves in isogeny class 28561.a

Label	Equation
28561.a.371293.1	\(y^2 + x^3y = -2x^4 - 2x^3 + 2x^2 + 3x - 2\)

L-function data

Analytic rank:

\(1\)

Mordell-Weil rank:

\(1\)

Bad L-factors:

Prime	L-Factor
\(13\)	\( 1\)

Good L-factors:

Prime	L-Factor
\(2\)	\( ( 1 - 2 T + 2 T^{2} )( 1 + 2 T + 2 T^{2} )\)
\(3\)	\( 1 + 3 T + 5 T^{2} + 9 T^{3} + 9 T^{4}\)
\(5\)	\( 1 + 25 T^{4}\)
\(7\)	\( 1 + 49 T^{4}\)
\(11\)	\( 1 + 121 T^{4}\)
\(17\)	\( ( 1 + 17 T^{2} )^{2}\)
\(19\)	\( 1 + 361 T^{4}\)
\(23\)	\( ( 1 + 23 T^{2} )^{2}\)
\(29\)	\( 1 + 3 T + 31 T^{2} + 87 T^{3} + 841 T^{4}\)
$\cdots$	$\cdots$

See L-function page for more information

Sato-Tate group

\(\mathrm{ST} =\) $F_{ac}$, \(\quad \mathrm{ST}^0 = \mathrm{U}(1)\times\mathrm{U}(1)\)

Decomposition of the Jacobian

Simple over \(\overline{\Q}\)

Endomorphisms of the Jacobian

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

Endomorphism algebra over \(\Q\):

\(\End (J_{}) \otimes \Q \)	\(\simeq\)	\(\Q\)
\(\End (J_{}) \otimes \R\)	\(\simeq\)	\(\R\)

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) 4.0.2197.1 with defining polynomial \(x^{4} - x^{3} + 2 x^{2} + 4 x + 3\)

Endomorphism algebra over \(\overline{\Q}\):

\(\End (J_{\overline{\Q}}) \otimes \Q \)	\(\simeq\)	4.0.2197.1 (CM)
\(\End (J_{\overline{\Q}}) \otimes \R\)	\(\simeq\)	\(\C \times \C\)

More complete information on endomorphism algebras and rings can be found on the pages of the individual curves in the isogeny class.