Genus 2 isogeny class 180625.a

• Label: 180625.a
• Conductor: $180625$
• Sato-Tate group: $N(\mathrm{SU}(2)\times\mathrm{SU}(2))$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\R \times \R\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathsf{RM}\)
• \(\End(J) \otimes \Q\): \(\Q\)
• \(\overline{\Q}\)-simple: yes
• \(\mathrm{GL}_2\)-type: no

L-function data

Analytic rank:

\(1\)

Mordell-Weil rank:

\(1\)

Bad L-factors:

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

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(2\)	\( 1 + 2 T^{2} + 4 T^{4}\)	2.2.a_c
\(3\)	\( 1 - T^{2} + 9 T^{4}\)	2.3.a_ab
\(7\)	\( 1 - 8 T^{2} + 49 T^{4}\)	2.7.a_ai
\(11\)	\( 1 + 5 T + 27 T^{2} + 55 T^{3} + 121 T^{4}\)	2.11.f_bb
\(13\)	\( 1 + 13 T^{2} + 169 T^{4}\)	2.13.a_n
\(19\)	\( 1 - 3 T + 39 T^{2} - 57 T^{3} + 361 T^{4}\)	2.19.ad_bn
\(23\)	\( ( 1 - 6 T + 23 T^{2} )( 1 + 6 T + 23 T^{2} )\)	2.23.a_k
\(29\)	\( 1 - 8 T + 69 T^{2} - 232 T^{3} + 841 T^{4}\)	2.29.ai_cr
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Sato-Tate group

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

Decomposition of the Jacobian

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

Endomorphisms of the Jacobian

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 \(K = \Q (a) \simeq \) \(\Q(\sqrt{5}) \) with defining polynomial \(x^{2} - x - 1\)

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

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

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

Genus 2 curves in isogeny class 180625.a

Label	Equation
180625.a.903125.1	\(y^2 + x^3y = x^5 + x - 1\)