Genus 2 Isogeny Class 7225.a

• Label: 7225.a
• Sato-Tate group: $G_{3,3}$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\R \times \R\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathrm{RM}\)
• \(\overline{\Q}\)-simple: yes
• \(\mathrm{GL}_2\)-type: yes

Genus 2 curves in isogeny class 7225.a

Label	Equation
7225.a.36125.1	\(y^2 + (x^3 + 1)y = -x^5 + 2x^4 - 3x^2 - x\)

L-function data

Analytic rank:

\(2\)  (upper bound)

Mordell-Weil rank:

\(2\)

Bad L-factors:

Prime	L-Factor
\(5\)	\( ( 1 + T )^{2}\)
\(17\)	\( ( 1 + T )^{2}\)

Good L-factors:

Prime	L-Factor
\(2\)	\( 1 + 2 T + 3 T^{2} + 4 T^{3} + 4 T^{4}\)
\(3\)	\( 1 + 4 T + 8 T^{2} + 12 T^{3} + 9 T^{4}\)
\(7\)	\( 1 + 4 T + 16 T^{2} + 28 T^{3} + 49 T^{4}\)
\(11\)	\( 1 + 8 T + 36 T^{2} + 88 T^{3} + 121 T^{4}\)
\(13\)	\( 1 + 18 T^{2} + 169 T^{4}\)
\(19\)	\( 1 + 30 T^{2} + 361 T^{4}\)
\(23\)	\( 1 + 4 T + 48 T^{2} + 92 T^{3} + 529 T^{4}\)
\(29\)	\( 1 + 4 T + 54 T^{2} + 116 T^{3} + 841 T^{4}\)
$\cdots$	$\cdots$

See L-function page for more information

Sato-Tate group

\(\mathrm{ST} =\) $G_{3,3}$, \(\quad \mathrm{ST}^0 = \mathrm{SU}(2)\times\mathrm{SU}(2)\)

Endomorphisms of the Jacobian

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

Endomorphism algebra over \(\Q\):

\(\End (J_{}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{2}) \)
\(\End (J_{}) \otimes \R\)	\(\simeq\)	\(\R \times \R\)

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.