Genus 2 isogeny class 188356.b

• Label: 188356.b
• Conductor: $188356$
• 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

L-function data

Analytic rank:

\(2\)  (upper bound)

Mordell-Weil rank:

\(2\)

Bad L-factors:

Prime	L-Factor
\(2\)	\( 1 + T + T^{2}\)
\(7\)	\( 1 - T + 7 T^{2}\)
\(31\)	\( 1 + 11 T + 31 T^{2}\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(3\)	\( ( 1 + 2 T + 3 T^{2} )^{2}\)	2.3.e_k
\(5\)	\( 1 + 2 T^{2} + 25 T^{4}\)	2.5.a_c
\(11\)	\( 1 - 6 T + 23 T^{2} - 66 T^{3} + 121 T^{4}\)	2.11.ag_x
\(13\)	\( 1 + 4 T + 3 T^{2} + 52 T^{3} + 169 T^{4}\)	2.13.e_d
\(17\)	\( ( 1 + 3 T + 17 T^{2} )^{2}\)	2.17.g_br
\(19\)	\( 1 + 12 T + 67 T^{2} + 228 T^{3} + 361 T^{4}\)	2.19.m_cp
\(23\)	\( 1 + 9 T + 50 T^{2} + 207 T^{3} + 529 T^{4}\)	2.23.j_by
\(29\)	\( 1 - 10 T^{2} + 841 T^{4}\)	2.29.a_ak
$\cdots$	$\cdots$	$\cdots$

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 \(\Q (b) \simeq \) 6.6.481170140857.2 with defining polynomial:
  \(x^{6} - x^{5} - 90 x^{4} - 197 x^{3} + 974 x^{2} + 2568 x - 216\)

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
  \(g_4 = \frac{822572}{4131} b^{5} - \frac{24071329}{132192} b^{4} - \frac{790588679}{44064} b^{3} - \frac{336421157}{8262} b^{2} + \frac{25207787687}{132192} b + \frac{5807503411}{11016}\)
  \(g_6 = -\frac{690048306955}{4758912} b^{5} + \frac{39620662249}{297432} b^{4} + \frac{767341747879}{58752} b^{3} + \frac{141002694663803}{4758912} b^{2} - \frac{660578532204017}{4758912} b - \frac{304311882533645}{793152}\)
   Conductor norm: 64

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 \(K = \Q (a) \simeq \) 6.6.481170140857.2 with defining polynomial \(x^{6} - x^{5} - 90 x^{4} - 197 x^{3} + 974 x^{2} + 2568 x - 216\)

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 188356.b

Label	Equation
188356.b.376712.1	\(y^2 + (x^3 + x + 1)y = -4x^4 - 5x^3 + x\)