Genus 2 isogeny class 3969.d

• Label: 3969.d
• Conductor: $3969$
• Sato-Tate group: $J(E_1)$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\mathrm{M}_2(\R)\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathrm{M}_2(\Q)\)
• \(\End(J) \otimes \Q\): \(\mathsf{RM}\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: yes

L-function data

Analytic rank:

\(0\)

Mordell-Weil rank:

\(0\)

Bad L-factors:

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

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(2\)	\( 1 + T^{2} + 4 T^{4}\)	2.2.a_b
\(5\)	\( 1 - 2 T^{2} + 25 T^{4}\)	2.5.a_ac
\(11\)	\( 1 + 10 T^{2} + 121 T^{4}\)	2.11.a_k
\(13\)	\( ( 1 - 2 T + 13 T^{2} )^{2}\)	2.13.ae_be
\(17\)	\( 1 + 22 T^{2} + 289 T^{4}\)	2.17.a_w
\(19\)	\( ( 1 + 4 T + 19 T^{2} )^{2}\)	2.19.i_cc
\(23\)	\( 1 + 34 T^{2} + 529 T^{4}\)	2.23.a_bi
\(29\)	\( ( 1 + 29 T^{2} )^{2}\)	2.29.a_cg
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Sato-Tate group

\(\mathrm{ST} =\) $J(E_1)$, \(\quad \mathrm{ST}^0 = \mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) \(\Q(\sqrt{-3}) \) with defining polynomial:
  \(x^{2} - x + 1\)

Decomposes up to isogeny as the square of the elliptic curve isogeny class:
  Elliptic curve isogeny class 2.0.3.1-441.2-a

Endomorphisms of the Jacobian

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

Endomorphism algebra over \(\Q\):

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

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\sqrt{-3}) \) with defining polynomial \(x^{2} - x + 1\)

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 3969.d

Label	Equation
3969.d.250047.1	\(y^2 + (x^2 + x + 1)y = -3x^5 + 5x^4 - 4x^3 + x\)