Genus 2 isogeny class 82369.a

• Label: 82369.a
• Conductor: $82369$
• Sato-Tate group: $\mathrm{SU}(2)\times\mathrm{SU}(2)$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\R \times \R\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathsf{RM}\)
• \(\End(J) \otimes \Q\): \(\mathsf{RM}\)
• \(\overline{\Q}\)-simple: yes
• \(\mathrm{GL}_2\)-type: yes

Genus 2 curves in isogeny class 82369.a

Label	Equation
82369.a.82369.1	\(y^2 + (x^3 + x + 1)y = -x^5 + x^3 - 4x^2 + 3x - 2\)

L-function data

Analytic rank:

\(2\)  (upper bound)

Mordell-Weil rank:

\(2\)

Bad L-factors:

Prime	L-Factor
\(7\)	\( ( 1 + T )^{2}\)
\(41\)	\( ( 1 + T )^{2}\)

Good L-factors:

Prime	L-Factor
\(2\)	\( 1 + T + 3 T^{2} + 2 T^{3} + 4 T^{4}\)
\(3\)	\( 1 + T + 5 T^{2} + 3 T^{3} + 9 T^{4}\)
\(5\)	\( 1 - T + 9 T^{2} - 5 T^{3} + 25 T^{4}\)
\(11\)	\( ( 1 + T + 11 T^{2} )^{2}\)
\(13\)	\( 1 + 8 T + 37 T^{2} + 104 T^{3} + 169 T^{4}\)
\(17\)	\( 1 + 4 T + 33 T^{2} + 68 T^{3} + 289 T^{4}\)
\(19\)	\( 1 + T + 27 T^{2} + 19 T^{3} + 361 T^{4}\)
\(23\)	\( 1 + 5 T + 51 T^{2} + 115 T^{3} + 529 T^{4}\)
\(29\)	\( 1 + 5 T + 53 T^{2} + 145 T^{3} + 841 T^{4}\)
$\cdots$	$\cdots$

See L-function page for more information

Sato-Tate group

\(\mathrm{ST} =\) $\mathrm{SU}(2)\times\mathrm{SU}(2)$, \(\quad \mathrm{ST}^0 = \mathrm{SU}(2)\times\mathrm{SU}(2)\)

Decomposition of the Jacobian

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

Endomorphisms of the Jacobian

Of \(\GL_2\)-type over \(\Q\)

Endomorphism algebra over \(\Q\):

\(\End (J_{}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{5}) \)
\(\End (J_{}) \otimes \R\)	\(\simeq\)	\(\R \times \R\)

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.