Abelian variety isogeny class 2.16.aj_bs over $\F_{2^{4}}$

• Label: 2.16.aj_bs
• Base field: $\F_{2^{4}}$
• Dimension: $2$
• $p$-rank: $1$
• Ordinary: no
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{2^{4}}$
Dimension:	$2$
L-polynomial:	$1 - 9 x + 44 x^{2} - 144 x^{3} + 256 x^{4}$
Frobenius angles:	$\pm0.126935807746$, $\pm0.434779740724$
Angle rank:	$2$ (numerical)
Number field:	4.0.40293.1
Galois group:	$D_{4}$
Jacobians:	$4$

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$148$	$67192$	$16892572$	$4273008048$	$1098559096948$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$8$	$264$	$4124$	$65200$	$1047668$	$16785960$	$268498700$	$4295107168$	$68719444580$	$1099512052824$

Jacobians and polarizations

This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2+xy=ax^5+(a^3+a)x^3+x$
• $y^2+xy=a^2x^5+a^3x^3+x$
• $y^2+xy=(a+1)x^5+(a^3+a^2)x^3+x$
• $y^2+xy=(a^2+1)x^5+(a^3+a^2+a+1)x^3+x$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{4}}$.

Endomorphism algebra over $\F_{2^{4}}$

The endomorphism algebra of this simple isogeny class is 4.0.40293.1.

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
2.16.j_bs	$2$	2.256.h_afo