Genus 2 isogeny class 278784.b

• Label: 278784.b
• Conductor: $278784$
• Sato-Tate group: $J(E_1)$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\mathrm{M}_2(\R)\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathrm{M}_2(\Q)\)
• \(\End(J) \otimes \Q\): \(\Q \times \Q\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: yes

L-function data

Analytic rank:

\(1\)

Mordell-Weil rank:

\(1\)

Bad L-factors:

Prime	L-Factor
\(2\)	\( 1\)
\(3\)	\( ( 1 - T )( 1 + T )\)
\(11\)	\( ( 1 - T )( 1 + T )\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(5\)	\( ( 1 - 2 T + 5 T^{2} )( 1 + 2 T + 5 T^{2} )\)	2.5.a_g
\(7\)	\( ( 1 + 2 T + 7 T^{2} )^{2}\)	2.7.e_s
\(13\)	\( ( 1 - 2 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} )\)	2.13.a_w
\(17\)	\( ( 1 - 4 T + 17 T^{2} )^{2}\)	2.17.ai_by
\(19\)	\( ( 1 - 6 T + 19 T^{2} )( 1 + 6 T + 19 T^{2} )\)	2.19.a_c
\(23\)	\( ( 1 + 23 T^{2} )^{2}\)	2.23.a_bu
\(29\)	\( ( 1 - 8 T + 29 T^{2} )( 1 + 8 T + 29 T^{2} )\)	2.29.a_ag
$\cdots$	$\cdots$	$\cdots$

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 \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
  Elliptic curve isogeny class 132.b
  Elliptic curve isogeny class 2112.c

Endomorphisms of the Jacobian

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

Endomorphism algebra over \(\Q\):

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

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

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 278784.b

Label	Equation
278784.b.557568.1	\(y^2 + y = -6x^6 - 8x^4 - 4x^2 - 1\)