Genus 2 isogeny class 147456.f

• Label: 147456.f
• Conductor: $147456$
• Sato-Tate group: $\mathrm{SU}(2)\times\mathrm{SU}(2)$
• End⁡(JQ‾)⊗R\End(J_{\overline{\Q}}) \otimes \R: $\R \times \R$
• End⁡(JQ‾)⊗Q\End(J_{\overline{\Q}}) \otimes \Q: $\Q \times \Q$
• 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 147456.f

Label	Equation
147456.f.884736.1	$y^2 = 2x^5 - 7x^3 + x^2 + 6x - 2$

L-function data

Analytic rank:

$0$

Mordell-Weil rank:

$0$

Bad L-factors:

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

Good L-factors:

Prime	L-Factor
$5$	$( 1 - 2 T + 5 T^{2} )^{2}$
$7$	$( 1 - 4 T + 7 T^{2} )( 1 + 2 T + 7 T^{2} )$
$11$	$( 1 - 2 T + 11 T^{2} )( 1 + 4 T + 11 T^{2} )$
$13$	$( 1 - 2 T + 13 T^{2} )^{2}$
$17$	$( 1 - 4 T + 17 T^{2} )( 1 + 2 T + 17 T^{2} )$
$19$	$( 1 - 4 T + 19 T^{2} )( 1 + 2 T + 19 T^{2} )$
$23$	$( 1 - 8 T + 23 T^{2} )( 1 + 4 T + 23 T^{2} )$
$29$	$( 1 - 6 T + 29 T^{2} )( 1 + 6 T + 29 T^{2} )$
$\cdots$	$\cdots$

See L-function page for more information

Sato-Tate group

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

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 128.b
  Elliptic curve isogeny class 1152.o

Endomorphisms of the Jacobian

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

Endomorphism algebra over $\Q$:

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

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.