Genus 2 isogeny class 409600.a

• Label: 409600.a
• Conductor: $409600$
• 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 409600.a

Label	Equation
409600.a.409600.1	\(y^2 = -2x^6 - 14x^5 - 29x^4 - 7x^3 + 24x^2 + x - 3\)

L-function data

Analytic rank:

\(0\)

Mordell-Weil rank:

\(0\)

Bad L-factors:

Prime	L-Factor
\(2\)	\( 1\)
\(5\)	\( ( 1 - T )^{2}\)

Good L-factors:

Prime	L-Factor
\(3\)	\( 1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4}\)
\(7\)	\( 1 - 2 T + 10 T^{2} - 14 T^{3} + 49 T^{4}\)
\(11\)	\( ( 1 + 2 T + 11 T^{2} )^{2}\)
\(13\)	\( 1 + 6 T^{2} + 169 T^{4}\)
\(17\)	\( 1 + 14 T^{2} + 289 T^{4}\)
\(19\)	\( 1 + 18 T^{2} + 361 T^{4}\)
\(23\)	\( 1 - 14 T + 90 T^{2} - 322 T^{3} + 529 T^{4}\)
\(29\)	\( ( 1 - 2 T + 29 T^{2} )^{2}\)
$\cdots$	$\cdots$

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.