Abelian variety isogeny class 2.512.add_dxx over $\F_{2^{9}}$

• Label: 2.512.add_dxx
• Base field: $\F_{2^{9}}$
• 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:	$\F_{2^{9}}$
Dimension:	$2$
L-polynomial:	$1 - 81 x + 2649 x^{2} - 41472 x^{3} + 262144 x^{4}$
Frobenius angles:	$\pm0.0617802508693$, $\pm0.200203993815$
Angle rank:	$2$ (numerical)
Number field:	4.0.200937721.1
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$223241$	$68389210627$	$18012767893125716$	$4722369979363822208419$	$1237940220880324551921931601$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$432$	$260882$	$134205579$	$68719527618$	$35184377250072$	$18014398619961647$	$9223372037426692752$	$4722366482824018616386$	$2417851639227276411754203$	$1237940039285335359764035682$

Jacobians and polarizations

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{9}}$.

Endomorphism algebra over $\F_{2^{9}}$

The endomorphism algebra of this simple isogeny class is 4.0.200937721.1.

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
2.512.dd_dxx	$2$	(not in LMFDB)