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

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

Invariants

Base field:	$\F_{2^{10}}$
Dimension:	$2$
L-polynomial:	$1 - 121 x + 5697 x^{2} - 123904 x^{3} + 1048576 x^{4}$
Frobenius angles:	$\pm0.0214972246866$, $\pm0.148663638065$
Angle rank:	$2$ (numerical)
Number field:	4.0.387225.1
Galois group:	$D_{4}$

This isogeny class is simple and geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$930249$	$1096111466451$	$1152840695507517636$	$1208924273017241894269539$	$1267650581231639034226775345049$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$904$	$1045330$	$1073666563$	$1099510221154$	$1125899889970264$	$1152921504658285951$	$1180591620724388970616$	$1208925819614564822702914$	$1237940039285356801984184707$	$1267650600228227738510560145650$

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 endomorphism algebra of this simple isogeny class is 4.0.387225.1.

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.er_ild	$2$	(not in LMFDB)