Abelian variety isogeny class 2.1024.aeq_iix over $\F_{2^{10}}$

• Label: 2.1024.aeq_iix
• Base field: $\F_{2^{10}}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{2^{10}}$
Dimension:	$2$
L-polynomial:	$( 1 - 63 x + 1024 x^{2} )( 1 - 57 x + 1024 x^{2} )$
	$1 - 120 x + 5639 x^{2} - 122880 x^{3} + 1048576 x^{4}$
Frobenius angles:	$\pm0.0563432964760$, $\pm0.150267280813$
Angle rank:	$2$ (numerical)

This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$2$
Slopes:	$[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$931216$	$1096242374656$	$1152849996118925104$	$1208924787902243336331264$	$1267650606031793288026545428176$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$905$	$1045455$	$1073675225$	$1099510689439$	$1125899911997225$	$1152921505606533231$	$1180591620762839052665$	$1208925819616060589783359$	$1237940039285413314087463625$	$1267650600228229828364496840975$

Jacobians and polarizations

This isogeny class contains a Jacobian, and hence is principally polarizable.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{10}}$.

Endomorphism algebra over $\F_{2^{10}}$

The isogeny class factors as 1.1024.acl $\times$ 1.1024.acf and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.1024.acl : \(\Q(\sqrt{-127}) \).
• 1.1024.acf : \(\Q(\sqrt{-7}) \).

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.1024.ag_achj	$2$	(not in LMFDB)
2.1024.g_achj	$2$	(not in LMFDB)
2.1024.eq_iix	$2$	(not in LMFDB)