Genus 2 isogeny class 324.a

• Label: 324.a
• Conductor: $324$
• Sato-Tate group: $E_3$
• End⁡(JQ‾)⊗R\End(J_{\overline{\Q}}) \otimes \R: $\mathrm{M}_2(\R)$
• End⁡(JQ‾)⊗Q\End(J_{\overline{\Q}}) \otimes \Q: $\mathrm{M}_2(\Q)$
• End⁡(J)⊗Q\End(J) \otimes \Q: $\mathsf{CM}$
• Q‾\overline{\Q}-simple: no
• GL2\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	Isogeny Class over $\mathbb{F}_p$
$5$	$1 - 5 T^{2} + 25 T^{4}$	2.5.a_af
$7$	$1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4}$	2.7.c_ad
$11$	$1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4}$	2.11.ad_ac
$13$	$( 1 - 5 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} )$	2.13.c_aj
$17$	$( 1 + 3 T + 17 T^{2} )^{2}$	2.17.g_br
$19$	$( 1 + T + 19 T^{2} )^{2}$	2.19.c_bn
$23$	$1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4}$	2.23.ag_n
$29$	$1 + 6 T + 7 T^{2} + 174 T^{3} + 841 T^{4}$	2.29.g_h
$\cdots$	$\cdots$	$\cdots$

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.