Genus 2 isogeny class 7569.a

• Label: 7569.a
• Conductor: $7569$
• 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 7569.a

Label	Equation
7569.a.68121.1	\(y^2 + (x^3 + x + 1)y = -x^4 + x^3 - 3x^2 + x - 1\)
7569.a.613089.1	\(y^2 + (x^3 + x + 1)y = -x^6 + 3x^5 - 10x^4 + 11x^3 - 15x^2 + 7x - 5\)

L-function data

Analytic rank:

\(0\)

Mordell-Weil rank:

\(0\)

Bad L-factors:

Prime	L-Factor
\(3\)	\( ( 1 - T )^{2}\)
\(29\)	\( ( 1 + T )^{2}\)

Good L-factors:

Prime	L-Factor
\(2\)	\( 1 - T + 3 T^{2} - 2 T^{3} + 4 T^{4}\)
\(5\)	\( 1 - 2 T + 6 T^{2} - 10 T^{3} + 25 T^{4}\)
\(7\)	\( 1 + 4 T + 13 T^{2} + 28 T^{3} + 49 T^{4}\)
\(11\)	\( 1 - 4 T + 21 T^{2} - 44 T^{3} + 121 T^{4}\)
\(13\)	\( 1 + 2 T + 7 T^{2} + 26 T^{3} + 169 T^{4}\)
\(17\)	\( ( 1 - 3 T + 17 T^{2} )^{2}\)
\(19\)	\( 1 + 10 T + 58 T^{2} + 190 T^{3} + 361 T^{4}\)
\(23\)	\( 1 + 2 T + 2 T^{2} + 46 T^{3} + 529 T^{4}\)
$\cdots$	$\cdots$

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.