Genus 2 isogeny class 692224.a

• Label: 692224.a
• Conductor: $692224$
• Sato-Tate group: $J(E_2)$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\mathrm{M}_2(\R)\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathrm{M}_2(\Q)\)
• \(\End(J) \otimes \Q\): \(\Q\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: no

Genus 2 curves in isogeny class 692224.a

Label	Equation
692224.a.692224.1	\(y^2 = x^5 - 3x^3 - x\)

L-function data

Analytic rank:

\(0\)

Mordell-Weil rank:

\(0\)

Bad L-factors:

Prime	L-Factor
\(2\)	\( 1\)
\(13\)	\( 1 + T^{2}\)

Good L-factors:

Prime	L-Factor
\(3\)	\( 1 - 2 T^{2} + 9 T^{4}\)
\(5\)	\( ( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} )\)
\(7\)	\( 1 - 4 T^{2} + 49 T^{4}\)
\(11\)	\( 1 + 20 T^{2} + 121 T^{4}\)
\(17\)	\( ( 1 - 4 T + 17 T^{2} )^{2}\)
\(19\)	\( 1 + 36 T^{2} + 361 T^{4}\)
\(23\)	\( 1 + 14 T^{2} + 529 T^{4}\)
\(29\)	\( 1 - 54 T^{2} + 841 T^{4}\)
$\cdots$	$\cdots$

See L-function page for more information

Sato-Tate group

\(\mathrm{ST} =\) $J(E_2)$, \(\quad \mathrm{ST}^0 = \mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) \(\Q(\zeta_{8})\) with defining polynomial:
  \(x^{4} + 1\)

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
  \(g_4 = -\frac{28160}{28561} b^{2} + \frac{581376}{28561}\)
  \(g_6 = -\frac{416530432}{4826809} b^{3} + \frac{355147776}{4826809} b\)
   Conductor norm: 7311616

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 \(K = \Q (a) \simeq \) \(\Q(\zeta_{8})\) with defining polynomial \(x^{4} + 1\)

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.