Genus 2 isogeny class 102400.e

• Label: 102400.e
• Conductor: $102400$
• Sato-Tate group: $J(E_1)$
• 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: $\Q \times \Q$
• Q‾\overline{\Q}-simple: no
• GL2\mathrm{GL}_2-type: yes

Genus 2 curves in isogeny class 102400.e

Label	Equation
102400.e.102400.1	$y^2 = x^5 + 3x^3 + x$

L-function data

Analytic rank:

$0$

Mordell-Weil rank:

$0$

Bad L-factors:

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

Good L-factors:

Prime	L-Factor
$3$	$( 1 - 2 T + 3 T^{2} )( 1 + 2 T + 3 T^{2} )$
$7$	$( 1 - 2 T + 7 T^{2} )( 1 + 2 T + 7 T^{2} )$
$11$	$( 1 - 4 T + 11 T^{2} )( 1 + 4 T + 11 T^{2} )$
$13$	$( 1 - 6 T + 13 T^{2} )^{2}$
$17$	$( 1 - 2 T + 17 T^{2} )^{2}$
$19$	$( 1 - 8 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$
$23$	$( 1 - 6 T + 23 T^{2} )( 1 + 6 T + 23 T^{2} )$
$29$	$( 1 - 2 T + 29 T^{2} )^{2}$
$\cdots$	$\cdots$

See L-function page for more information

Sato-Tate group

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

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 320.e
  Elliptic curve isogeny class 320.b

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 $K = \Q (a) \simeq$ $\Q(\sqrt{-1})$ with defining polynomial $x^{2} + 1$

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.