Genus 2 isogeny class 537289.c

• Label: 537289.c
• Conductor: $537289$
• 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
\(733\)	\( 1 + 50 T + 733 T^{2}\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(2\)	\( 1 - T^{2} + 4 T^{4}\)	2.2.a_ab
\(3\)	\( 1 + 2 T + T^{2} + 6 T^{3} + 9 T^{4}\)	2.3.c_b
\(5\)	\( 1 - 7 T^{2} + 25 T^{4}\)	2.5.a_ah
\(7\)	\( ( 1 + T + 7 T^{2} )( 1 + 5 T + 7 T^{2} )\)	2.7.g_t
\(11\)	\( 1 - 10 T^{2} + 121 T^{4}\)	2.11.a_ak
\(13\)	\( 1 + 13 T^{2} + 169 T^{4}\)	2.13.a_n
\(17\)	\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4}\)	2.17.d_ai
\(19\)	\( ( 1 + 2 T + 19 T^{2} )^{2}\)	2.19.e_bq
\(23\)	\( 1 + 12 T + 71 T^{2} + 276 T^{3} + 529 T^{4}\)	2.23.m_ct
\(29\)	\( ( 1 - 29 T^{2} )^{2}\)	2.29.a_acg
$\cdots$	$\cdots$	$\cdots$

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 \(\Q (b)\) with defining polynomial:
  \(x^{6} - x^{5} - 305 x^{4} - 3190 x^{3} - 13345 x^{2} - 24521 x - 15791\)

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
  \(g_4 = -\frac{195527}{768} b^{5} + \frac{106703}{48} b^{4} + \frac{15517437}{256} b^{3} + \frac{262259057}{768} b^{2} + \frac{70172353}{96} b + \frac{392766647}{768}\)
  \(g_6 = \frac{148933139}{144} b^{5} - \frac{20301225907}{2304} b^{4} - \frac{191390106469}{768} b^{3} - \frac{410462090611}{288} b^{2} - \frac{2367815638657}{768} b - \frac{5002537483207}{2304}\)
   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 \(K = \Q (a)\) with defining polynomial \(x^{6} - x^{5} - 305 x^{4} - 3190 x^{3} - 13345 x^{2} - 24521 x - 15791\)

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 537289.c

Label	Equation
537289.c.537289.1	\(y^2 + (x^2 + x)y = x^6 - 2x^5 - 3x^4 + 5x^3 + 2x^2 - 4x + 1\)