Genus 2 isogeny class 76729.a

• Label: 76729.a
• Conductor: $76729$
• 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:

\(0\)

Mordell-Weil rank:

\(0\)

Bad L-factors:

Prime	L-Factor
\(277\)	\( 1 + 26 T + 277 T^{2}\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(2\)	\( 1 - T^{2} + 4 T^{4}\)	2.2.a_ab
\(3\)	\( 1 - T - 2 T^{2} - 3 T^{3} + 9 T^{4}\)	2.3.ab_ac
\(5\)	\( 1 - 3 T + 8 T^{2} - 15 T^{3} + 25 T^{4}\)	2.5.ad_i
\(7\)	\( ( 1 - 4 T + 7 T^{2} )( 1 + 5 T + 7 T^{2} )\)	2.7.b_ag
\(11\)	\( 1 + 3 T + 14 T^{2} + 33 T^{3} + 121 T^{4}\)	2.11.d_o
\(13\)	\( ( 1 - 2 T + 13 T^{2} )^{2}\)	2.13.ae_be
\(17\)	\( 1 - 3 T + 20 T^{2} - 51 T^{3} + 289 T^{4}\)	2.17.ad_u
\(19\)	\( ( 1 + 4 T + 19 T^{2} )^{2}\)	2.19.i_cc
\(23\)	\( 1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4}\)	2.23.ad_ao
\(29\)	\( 1 + 3 T - 20 T^{2} + 87 T^{3} + 841 T^{4}\)	2.29.d_au
$\cdots$	$\cdots$	$\cdots$

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 \(\Q (b) \simeq \) 6.6.1630793025157.1 with defining polynomial:
  \(x^{6} - x^{5} - 115 x^{4} - 590 x^{3} - 965 x^{2} - 305 x + 247\)

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
  \(g_4 = \frac{2200913}{4096} b^{5} - \frac{2470717}{2048} b^{4} - \frac{246994057}{4096} b^{3} - \frac{990865181}{4096} b^{2} - \frac{27672899}{128} b + \frac{447801295}{4096}\)
  \(g_6 = -\frac{759325973}{8192} b^{5} + \frac{3406852085}{16384} b^{4} + \frac{170401555667}{16384} b^{3} + \frac{171039056209}{4096} b^{2} + \frac{615693335017}{16384} b - \frac{299067260123}{16384}\)
   Conductor norm: 1

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 \(K = \Q (a) \simeq \) 6.6.1630793025157.1 with defining polynomial \(x^{6} - x^{5} - 115 x^{4} - 590 x^{3} - 965 x^{2} - 305 x + 247\)

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 76729.a

Label	Equation
76729.a.76729.1	\(y^2 + (x^2 + x)y = x^5 - 6x^4 + 6x^3 - x^2 - x\)