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

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

Invariants

Base field:	$\F_{2^{4}}$
Dimension:	$2$
L-polynomial:	$( 1 - 4 x )^{2}( 1 - 7 x + 16 x^{2} )$
	$1 - 15 x + 88 x^{2} - 240 x^{3} + 256 x^{4}$
Frobenius angles:	$0$, $0$, $\pm0.160861246510$
Angle rank:	$1$ (numerical)
Jacobians:	$0$

This isogeny class is not simple, 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$90$	$54000$	$16233210$	$4276044000$	$1099117082250$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$2$	$208$	$3962$	$65248$	$1048202$	$16777168$	$268432922$	$4294917568$	$68719038122$	$1099508828848$

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.ah 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.ah : \(\Q(\sqrt{-15}) \).

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.16.ab_ay	$2$	2.256.abx_boq
2.16.b_ay	$2$	2.256.abx_boq
2.16.p_dk	$2$	2.256.abx_boq
2.16.ad_e	$3$	(not in LMFDB)