Genus 2 isogeny class 37636.a

• Label: 37636.a
• Conductor: $37636$
• Sato-Tate group: $E_6$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\mathrm{M}_2(\R)\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathrm{M}_2(\Q)\)
• \(\End(J) \otimes \Q\): \(\mathsf{CM}\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: yes

L-function data

Analytic rank:

\(2\)  (upper bound)

Mordell-Weil rank:

\(2\)

Bad L-factors:

Prime	L-Factor
\(2\)	\( 1 + T + T^{2}\)
\(97\)	\( 1 + 14 T + 97 T^{2}\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(3\)	\( 1 + 2 T + T^{2} + 6 T^{3} + 9 T^{4}\)	2.3.c_b
\(5\)	\( 1 + 3 T + 8 T^{2} + 15 T^{3} + 25 T^{4}\)	2.5.d_i
\(7\)	\( ( 1 + T + 7 T^{2} )( 1 + 5 T + 7 T^{2} )\)	2.7.g_t
\(11\)	\( 1 - 11 T^{2} + 121 T^{4}\)	2.11.a_al
\(13\)	\( ( 1 + 2 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} )\)	2.13.j_bo
\(17\)	\( 1 - 3 T + 20 T^{2} - 51 T^{3} + 289 T^{4}\)	2.17.ad_u
\(19\)	\( ( 1 - 8 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )\)	2.19.a_aba
\(23\)	\( 1 + 23 T^{2} + 529 T^{4}\)	2.23.a_x
\(29\)	\( 1 + 12 T + 77 T^{2} + 348 T^{3} + 841 T^{4}\)	2.29.m_cz
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Sato-Tate group

\(\mathrm{ST} =\) $E_6$, \(\quad \mathrm{ST}^0 = \mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) 6.6.8587340257.1 with defining polynomial:
  \(x^{6} - x^{5} - 40 x^{4} - 45 x^{3} + 236 x^{2} + 230 x - 389\)

Decomposes up to isogeny as the square of the elliptic curve isogeny class:
  \(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
  \(g_4 = -\frac{780253}{376} b^{5} - \frac{3367987}{1504} b^{4} + \frac{117849811}{1504} b^{3} + \frac{385459869}{1504} b^{2} + \frac{16108321}{376} b - \frac{584419305}{1504}\)
  \(g_6 = \frac{14792680517}{6016} b^{5} + \frac{15949868983}{6016} b^{4} - \frac{558559142863}{6016} b^{3} - \frac{913242130137}{3008} b^{2} - \frac{304797991689}{6016} b + \frac{173053323651}{376}\)
   Conductor norm: 64

Endomorphisms of the Jacobian

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

Endomorphism algebra over \(\Q\):

\(\End (J_{}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{-3}) \)
\(\End (J_{}) \otimes \R\)	\(\simeq\)	\(\C\)

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) 6.6.8587340257.1 with defining polynomial \(x^{6} - x^{5} - 40 x^{4} - 45 x^{3} + 236 x^{2} + 230 x - 389\)

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

\(\End (J_{\overline{\Q}}) \otimes \Q \)	\(\simeq\)	\(\mathrm{M}_2(\)\(\Q\)\()\)
\(\End (J_{\overline{\Q}}) \otimes \R\)	\(\simeq\)	\(\mathrm{M}_2 (\R)\)

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 37636.a

Label	Equation
37636.a.602176.1	\(y^2 + (x^3 + x + 1)y = x^5 + 4x^4 + 6x^3 + 3x^2\)