Genus 2 isogeny class 324.a

• Label: 324.a
• Conductor: $324$
• Sato-Tate group: $E_3$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\mathrm{M}_2(\R)\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathrm{M}_2(\Q)\)
• \(\End(J) \otimes \Q\): \(\mathsf{CM}\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: yes

Genus 2 curves in isogeny class 324.a

Label	Equation
324.a.648.1	\(y^2 + (x^3 + x + 1)y = x^5 + 2x^4 + 2x^3 + x^2\)

L-function data

Analytic rank:

\(0\)

Mordell-Weil rank:

\(0\)

Bad L-factors:

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

Good L-factors:

Prime	L-Factor
\(5\)	\( 1 - 5 T^{2} + 25 T^{4}\)
\(7\)	\( 1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4}\)
\(11\)	\( 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4}\)
\(13\)	\( ( 1 - 5 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} )\)
\(17\)	\( ( 1 + 3 T + 17 T^{2} )^{2}\)
\(19\)	\( ( 1 + T + 19 T^{2} )^{2}\)
\(23\)	\( 1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4}\)
\(29\)	\( 1 + 6 T + 7 T^{2} + 174 T^{3} + 841 T^{4}\)
$\cdots$	$\cdots$

See L-function page for more information

Sato-Tate group

\(\mathrm{ST} =\) $E_3$, \(\quad \mathrm{ST}^0 = \mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) \(\Q(\zeta_{9})^+\) with defining polynomial:
  \(x^{3} - 3 x - 1\)

Decomposes up to isogeny as the square of the elliptic curve isogeny class:
  Elliptic curve isogeny class 3.3.81.1-8.1-a

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 \(K = \Q (a) \simeq \) \(\Q(\zeta_{9})^+\) with defining polynomial \(x^{3} - 3 x - 1\)

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.