Genus 2 isogeny class 15876.b

• Label: 15876.b
• Conductor: $15876$
• Sato-Tate group: $E_3$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\mathrm{M}_2(\R)\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathrm{M}_2(\Q)\)
• \(\End(J) \otimes \Q\): \(\mathsf{CM}\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: yes

L-function data

Analytic rank:

\(0\)

Mordell-Weil rank:

\(0\)

Bad L-factors:

Prime	L-Factor
\(2\)	\( 1 - T + T^{2}\)
\(3\)	\( 1 + 3 T^{2}\)
\(7\)	\( 1 + T + T^{2}\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(5\)	\( 1 - 3 T + 4 T^{2} - 15 T^{3} + 25 T^{4}\)	2.5.ad_e
\(11\)	\( 1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4}\)	2.11.ag_z
\(13\)	\( ( 1 - 5 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} )\)	2.13.c_aj
\(17\)	\( ( 1 - 6 T + 17 T^{2} )^{2}\)	2.17.am_cs
\(19\)	\( ( 1 + 7 T + 19 T^{2} )^{2}\)	2.19.o_dj
\(23\)	\( 1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4}\)	2.23.d_ao
\(29\)	\( 1 + 6 T + 7 T^{2} + 174 T^{3} + 841 T^{4}\)	2.29.g_h
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Sato-Tate group

\(\mathrm{ST} =\) $E_3$, \(\quad \mathrm{ST}^0 = \mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) \(\Q(\zeta_{9})^+\) with defining polynomial:
  \(x^{3} - 3 x - 1\)

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
  \(g_4 = \frac{41553}{16} b^{2} + \frac{78489}{16} b + \frac{23085}{16}\)
  \(g_6 = \frac{29032425}{64} b^{2} + \frac{54580959}{64} b + \frac{1935495}{8}\)
   Conductor norm: 2744

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 \(K = \Q (a) \simeq \) \(\Q(\zeta_{9})^+\) with defining polynomial \(x^{3} - 3 x - 1\)

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.

Genus 2 curves in isogeny class 15876.b

Label	Equation
15876.b.222264.1	\(y^2 + (x^3 + x + 1)y = -x^6 + 2x^5 - 4x^4 + 4x^3 - 5x^2 + 2x - 1\)