Genus 2 isogeny class 6272.b

• Label: 6272.b
• Conductor: $6272$
• Sato-Tate group: $\mathrm{SU}(2)\times\mathrm{SU}(2)$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\R \times \R\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\Q \times \Q\)
• \(\End(J) \otimes \Q\): \(\Q \times \Q\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: yes

L-function data

Analytic rank:

\(1\)

Mordell-Weil rank:

\(1\)

Bad L-factors:

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

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(3\)	\( ( 1 + 3 T^{2} )( 1 + 2 T + 3 T^{2} )\)	2.3.c_g
\(5\)	\( ( 1 + 5 T^{2} )( 1 + 2 T + 5 T^{2} )\)	2.5.c_k
\(11\)	\( ( 1 + 11 T^{2} )( 1 + 4 T + 11 T^{2} )\)	2.11.e_w
\(13\)	\( ( 1 + 2 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} )\)	2.13.g_bi
\(17\)	\( ( 1 - 6 T + 17 T^{2} )( 1 + 6 T + 17 T^{2} )\)	2.17.a_ac
\(19\)	\( ( 1 - 8 T + 19 T^{2} )( 1 - 2 T + 19 T^{2} )\)	2.19.ak_cc
\(23\)	\( ( 1 + 23 T^{2} )^{2}\)	2.23.a_bu
\(29\)	\( ( 1 + 6 T + 29 T^{2} )^{2}\)	2.29.m_dq
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Sato-Tate group

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

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 448.e
  Elliptic curve isogeny class 14.a

Endomorphisms of the Jacobian

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

Endomorphism algebra over \(\Q\):

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

All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).

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 6272.b

Label	Equation
6272.b.50176.1	\(y^2 + xy = x^6 + 3x^4 + 3x^2 + 1\)