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

• Label: 3.16.at_gf_abfc
• 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:	$( 1 - 4 x )^{2}( 1 - 11 x + 57 x^{2} - 176 x^{3} + 256 x^{4} )$
	$1 - 19 x + 161 x^{2} - 808 x^{3} + 2576 x^{4} - 4864 x^{5} + 4096 x^{6}$
Frobenius angles:	$0$, $0$, $\pm0.0728689886706$, $\pm0.368631800070$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1143$	$14316075$	$66931501392$	$278133063001875$	$1147765479839878383$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-2$	$218$	$3991$	$64754$	$1043878$	$16760975$	$268411918$	$4294999394$	$68719483111$	$1099509752378$

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.al_cf 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.al_cf : 4.0.78057.3.

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.ad_ap_ea	$2$	(not in LMFDB)
3.16.d_ap_aea	$2$	(not in LMFDB)
3.16.t_gf_bfc	$2$	(not in LMFDB)
3.16.ah_bd_aeu	$3$	(not in LMFDB)