Genus 2 isogeny class 405769.b

• Label: 405769.b
• Conductor: $405769$
• 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

Genus 2 curves in isogeny class 405769.b

Label	Equation
405769.b.405769.1	\(y^2 + (x^3 + x + 1)y = -x^6 + 4x^5 - 5x^4 - 4x^3 + 4x^2 - 1\)

L-function data

Analytic rank:

\(0\)

Mordell-Weil rank:

\(0\)

Bad L-factors:

Prime	L-Factor
\(7\)	\( 1\)
\(13\)	\( 1 - 2 T + 13 T^{2}\)

Good L-factors:

Prime	L-Factor
\(2\)	\( 1 - 3 T + 5 T^{2} - 6 T^{3} + 4 T^{4}\)
\(3\)	\( 1 + T - 2 T^{2} + 3 T^{3} + 9 T^{4}\)
\(5\)	\( 1 - 7 T^{2} + 25 T^{4}\)
\(11\)	\( 1 - 9 T + 38 T^{2} - 99 T^{3} + 121 T^{4}\)
\(17\)	\( 1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4}\)
\(19\)	\( 1 + 3 T + 22 T^{2} + 57 T^{3} + 361 T^{4}\)
\(23\)	\( 1 - 23 T^{2} + 529 T^{4}\)
\(29\)	\( 1 + 3 T - 20 T^{2} + 87 T^{3} + 841 T^{4}\)
$\cdots$	$\cdots$

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 \(\Q (b) \simeq \) \(\Q(\zeta_{13})^+\) with defining polynomial:
  \(x^{6} - x^{5} - 5 x^{4} + 4 x^{3} + 6 x^{2} - 3 x - 1\)

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
  \(g_4 = -1843170 b^{5} - 921480 b^{4} + 7841323 b^{3} + 4388727 b^{2} - 4501063 b - 1225196\)
  \(g_6 = -5757093524 b^{5} - 2861644240 b^{4} + 24502003799 b^{3} + 13652376101 b^{2} - 14106174901 b - 3845778573\)
   Conductor norm: 13841287201

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 \(K = \Q (a) \simeq \) \(\Q(\zeta_{13})^+\) with defining polynomial \(x^{6} - x^{5} - 5 x^{4} + 4 x^{3} + 6 x^{2} - 3 x - 1\)

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.