Genus 2 isogeny class 104976.a

• Label: 104976.a
• Conductor: $104976$
• Sato-Tate group: $J(E_6)$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\mathrm{M}_2(\R)\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathrm{M}_2(\Q)\)
• \(\End(J) \otimes \Q\): \(\Q\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: no

Genus 2 curves in isogeny class 104976.a

Label	Equation
104976.a.104976.1	\(y^2 + y = -x^6 - 2x^3 - 1\)

L-function data

Analytic rank:

\(1\)

Mordell-Weil rank:

\(1\)

Bad L-factors:

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

Good L-factors:

Prime	L-Factor
\(5\)	\( 1 - 2 T^{2} + 25 T^{4}\)
\(7\)	\( ( 1 - 5 T + 7 T^{2} )( 1 + 5 T + 7 T^{2} )\)
\(11\)	\( ( 1 + 11 T^{2} )^{2}\)
\(13\)	\( 1 - T - 12 T^{2} - 13 T^{3} + 169 T^{4}\)
\(17\)	\( 1 + 22 T^{2} + 289 T^{4}\)
\(19\)	\( ( 1 + T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )\)
\(23\)	\( ( 1 - 6 T + 23 T^{2} )( 1 + 6 T + 23 T^{2} )\)
\(29\)	\( 1 + 10 T^{2} + 841 T^{4}\)
$\cdots$	$\cdots$

See L-function page for more information

Sato-Tate group

\(\mathrm{ST} =\) $J(E_6)$, \(\quad \mathrm{ST}^0 = \mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) 6.2.45349632.3 with defining polynomial:
  \(x^{6} - 18 x^{3} + 6\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
  \(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
  \(g_4 = -\frac{558}{5} b^{5} + \frac{9882}{5} b^{2}\)
  \(g_6 = -\frac{9072}{5} b^{3} + \frac{148068}{5}\)
   Conductor norm: 81
  \(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
  \(g_4 = \frac{18}{5} b^{5} + \frac{378}{5} b^{2}\)
  \(g_6 = \frac{9072}{5} b^{3} - \frac{15228}{5}\)
   Conductor norm: 81

Endomorphisms of the Jacobian

Not of \(\GL_2\)-type over \(\Q\)

Endomorphism algebra over \(\Q\):

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

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a)\) with defining polynomial \(x^{12} + 36\)

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.