Abelian variety isogeny class 3.16.ar_fj_abau over $\F_{2^{4}}$

• Label: 3.16.ar_fj_abau
• Base field: $\F_{2^{4}}$
• Dimension: $3$
• $p$-rank: $2$
• 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:	$3$
L-polynomial:	$( 1 - 4 x )^{2}( 1 - 9 x + 51 x^{2} - 144 x^{3} + 256 x^{4} )$
	$1 - 17 x + 139 x^{2} - 696 x^{3} + 2224 x^{4} - 4352 x^{5} + 4096 x^{6}$
Frobenius angles:	$0$, $0$, $\pm0.252176979752$, $\pm0.361066333219$
Angle rank:	$2$ (numerical)

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1395$	$16077375$	$70178984820$	$281852927368875$	$1150364380978828125$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$0$	$246$	$4185$	$65626$	$1046250$	$16761591$	$268381260$	$4294828466$	$68718976425$	$1099508959446$

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 $\times$ 2.16.aj_bz 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 $\infty$.
• 2.16.aj_bz : 4.0.42625.1.

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
3.16.ab_af_eq	$2$	(not in LMFDB)
3.16.b_af_aeq	$2$	(not in LMFDB)
3.16.r_fj_bau	$2$	(not in LMFDB)
3.16.af_bf_adg	$3$	(not in LMFDB)