Genus 2 isogeny class 2304.b

• Label: 2304.b
• Conductor: $2304$
• Sato-Tate group: $E_1$
• 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: $\mathrm{M}_2(\Q)$
• Q‾\overline{\Q}-simple: no
• GL2\mathrm{GL}_2-type: no

Genus 2 curves in isogeny class 2304.b

Label	Equation
2304.b.147456.1	$y^2 = -x^6 - 2x^4 - 2x^2 - 1$

L-function data

Analytic rank:

$0$

Mordell-Weil rank:

$0$

Bad L-factors:

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

Good L-factors:

Prime	L-Factor	Isogeny Class over $\mathbb{F}_p$
$5$	$( 1 + 2 T + 5 T^{2} )^{2}$	2.5.e_o
$7$	$( 1 + 7 T^{2} )^{2}$	2.7.a_o
$11$	$( 1 + 4 T + 11 T^{2} )^{2}$	2.11.i_bm
$13$	$( 1 + 2 T + 13 T^{2} )^{2}$	2.13.e_be
$17$	$( 1 - 2 T + 17 T^{2} )^{2}$	2.17.ae_bm
$19$	$( 1 - 4 T + 19 T^{2} )^{2}$	2.19.ai_cc
$23$	$( 1 - 8 T + 23 T^{2} )^{2}$	2.23.aq_eg
$29$	$( 1 - 6 T + 29 T^{2} )^{2}$	2.29.am_dq
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Sato-Tate group

$\mathrm{ST} =$ $E_1$, $\quad \mathrm{ST}^0 = \mathrm{SU}(2)$

Decomposition of the Jacobian

Splits over $\Q$

Decomposes up to isogeny as the square of the elliptic curve isogeny class:
  Elliptic curve isogeny class 48.a

Endomorphisms of the Jacobian

Not of $\GL_2$-type over $\Q$

Endomorphism algebra over $\Q$:

$\End (J_{}) \otimes \Q$	$\simeq$	$\mathrm{M}_2($$\Q$$)$
$\End (J_{}) \otimes \R$	$\simeq$	$\mathrm{M}_2 (\R)$

All $\overline{\Q}$-endomorphisms of the Jacobian are defined over $\Q$.

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