Genus 2 isogeny class 2916.b

• Label: 2916.b
• Conductor: $2916$
• Sato-Tate group: $D_{3,2}$
• End⁡(JQ‾)⊗R\End(J_{\overline{\Q}}) \otimes \R: $\mathrm{M}_2(\C)$
• End⁡(JQ‾)⊗Q\End(J_{\overline{\Q}}) \otimes \Q: $\mathrm{M}_2(\mathsf{CM})$
• End⁡(J)⊗Q\End(J) \otimes \Q: $\Q \times \Q$
• Q‾\overline{\Q}-simple: no
• GL2\mathrm{GL}_2-type: yes

Genus 2 curves in isogeny class 2916.b

Label	Equation
2916.b.11664.1	$y^2 + y = x^6$

L-function data

Analytic rank:

$0$

Mordell-Weil rank:

$0$

Bad L-factors:

Prime	L-Factor
$2$	$1 + 2 T^{2}$
$3$	$1$

Good L-factors:

Prime	L-Factor
$5$	$( 1 + 5 T^{2} )^{2}$
$7$	$( 1 - 5 T + 7 T^{2} )( 1 + T + 7 T^{2} )$
$11$	$( 1 + 11 T^{2} )^{2}$
$13$	$( 1 - 5 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} )$
$17$	$( 1 + 17 T^{2} )^{2}$
$19$	$( 1 + T + 19 T^{2} )( 1 + 7 T + 19 T^{2} )$
$23$	$( 1 + 23 T^{2} )^{2}$
$29$	$( 1 + 29 T^{2} )^{2}$
$\cdots$	$\cdots$

See L-function page for more information

Sato-Tate group

$\mathrm{ST} =$ $D_{3,2}$, $\quad \mathrm{ST}^0 = \mathrm{U}(1)$

Decomposition of the Jacobian

Splits over $\Q$

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
  Elliptic curve isogeny class 108.a
  Elliptic curve isogeny class 27.a

Endomorphisms of the Jacobian

Of $\GL_2$-type over $\Q$

Endomorphism algebra over $\Q$:

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

Smallest field over which all endomorphisms are defined:
Galois number field $K = \Q (a) \simeq$ 6.0.34992.1 with defining polynomial $x^{6} - 3 x^{5} + 5 x^{3} - 3 x + 1$

Endomorphism algebra over $\overline{\Q}$:

$\End (J_{\overline{\Q}}) \otimes \Q$	$\simeq$	$\mathrm{M}_2($$\Q(\sqrt{-3})$$)$
$\End (J_{\overline{\Q}}) \otimes \R$	$\simeq$	$\mathrm{M}_2 (\C)$

More complete information on endomorphism algebras and rings can be found on the pages of the individual curves in the isogeny class.