Genus 2 isogeny class 40000.e

• Label: 40000.e
• Conductor: $40000$
• Sato-Tate group: $J(C_4)$
• 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: $\mathsf{CM}$
• Q‾\overline{\Q}-simple: no
• GL2\mathrm{GL}_2-type: yes

Genus 2 curves in isogeny class 40000.e

Label	Equation
40000.e.200000.1	$y^2 + x^3y = x^5 - 5x^3 - 10x^2 - 8x - 2$

L-function data

Analytic rank:

$2$  (upper bound)

Mordell-Weil rank:

$2$

Bad L-factors:

Prime	L-Factor
$2$	$1 + 2 T + 2 T^{2}$
$5$	$1$

Good L-factors:

Prime	L-Factor
$3$	$1 + 4 T + 8 T^{2} + 12 T^{3} + 9 T^{4}$
$7$	$1 + 49 T^{4}$
$11$	$( 1 + 6 T + 11 T^{2} )^{2}$
$13$	$1 + 169 T^{4}$
$17$	$1 + 8 T + 32 T^{2} + 136 T^{3} + 289 T^{4}$
$19$	$1 - 34 T^{2} + 361 T^{4}$
$23$	$1 + 529 T^{4}$
$29$	$( 1 + 29 T^{2} )^{2}$
$\cdots$	$\cdots$

See L-function page for more information

Sato-Tate group

$\mathrm{ST} =$ $J(C_4)$, $\quad \mathrm{ST}^0 = \mathrm{U}(1)$

Decomposition of the Jacobian

Splits over the number field $\Q (b) \simeq$ 4.4.8000.1 with defining polynomial:
  $x^{4} - 10 x^{2} + 20$

Decomposes up to isogeny as the square of the elliptic curve isogeny class:
  Elliptic curve isogeny class 4.4.8000.1-25.1-b

Endomorphisms of the Jacobian

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

Endomorphism algebra over $\Q$:

$\End (J_{}) \otimes \Q$	$\simeq$	$\Q(\sqrt{-1})$
$\End (J_{}) \otimes \R$	$\simeq$	$\C$

Smallest field over which all endomorphisms are defined:
Galois number field $K = \Q (a) \simeq$ 8.0.64000000.1 with defining polynomial $x^{8} - 2 x^{6} + 4 x^{4} - 8 x^{2} + 16$

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

$\End (J_{\overline{\Q}}) \otimes \Q$	$\simeq$	$\mathrm{M}_2($$\Q(\sqrt{-2})$$)$
$\End (J_{\overline{\Q}}) \otimes \R$	$\simeq$	$\mathrm{M}_2 (\C)$

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