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

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

Invariants

Base field:	$\F_{2^{4}}$
Dimension:	$2$
L-polynomial:	$( 1 - 4 x )^{2}( 1 - x + 16 x^{2} )$
	$1 - 9 x + 40 x^{2} - 144 x^{3} + 256 x^{4}$
Frobenius angles:	$0$, $0$, $\pm0.460106912325$
Angle rank:	$1$ (numerical)
Jacobians:	$0$

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$144$	$64800$	$16447536$	$4232347200$	$1096109357904$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$8$	$256$	$4016$	$64576$	$1045328$	$16775008$	$268427888$	$4294765696$	$68718478736$	$1099510185376$

Jacobians and polarizations

This isogeny class is principally polarizable, but does not contain a Jacobian.

Decomposition and endomorphism algebra

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

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

The isogeny class factors as 1.16.ai $\times$ 1.16.ab and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.16.ai : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• 1.16.ab : \(\Q(\sqrt{-7}) \).

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{4}}$.

Subfield	Primitive Model
$\F_{2}$	2.2.ab_e
$\F_{2}$	2.2.b_e

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.16.ah_y	$2$	2.256.ab_asm
2.16.h_y	$2$	2.256.ab_asm
2.16.j_bo	$2$	2.256.ab_asm
2.16.d_bc	$3$	(not in LMFDB)