Genus 2 isogeny class 12500.a

• Label: 12500.a
• Conductor: $12500$
• 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:

\(0\)

Mordell-Weil rank:

\(0\)

Bad L-factors:

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

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(3\)	\( 1 + 9 T^{4}\)	2.3.a_a
\(7\)	\( ( 1 - 2 T + 7 T^{2} )( 1 + 2 T + 7 T^{2} )\)	2.7.a_k
\(11\)	\( ( 1 - 2 T + 11 T^{2} )^{2}\)	2.11.ae_ba
\(13\)	\( 1 + 169 T^{4}\)	2.13.a_a
\(17\)	\( 1 - 20 T^{2} + 289 T^{4}\)	2.17.a_au
\(19\)	\( 1 - 5 T + 13 T^{2} - 95 T^{3} + 361 T^{4}\)	2.19.af_n
\(23\)	\( 1 - 30 T^{2} + 529 T^{4}\)	2.23.a_abe
\(29\)	\( 1 - 5 T + 33 T^{2} - 145 T^{3} + 841 T^{4}\)	2.29.af_bh
$\cdots$	$\cdots$	$\cdots$

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 12500.a

Label	Equation
12500.a.12500.1	\(y^2 + (x^3 + 1)y = x^6 + 2x^3 - x\)