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

• Label: 3.16.au_gx_abjk
• 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:	$( 1 - 4 x )^{2}( 1 - 7 x + 16 x^{2} )( 1 - 5 x + 16 x^{2} )$
	$1 - 20 x + 179 x^{2} - 920 x^{3} + 2864 x^{4} - 5120 x^{5} + 4096 x^{6}$
Frobenius angles:	$0$, $0$, $\pm0.160861246510$, $\pm0.285098958592$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1080$	$14256000$	$68374280520$	$282218904000000$	$1153030973368527000$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-3$	$215$	$4077$	$65711$	$1048677$	$16772135$	$268400157$	$4294834271$	$68719145877$	$1099510700375$

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 $\times$ 1.16.af 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 : \(\Q(\sqrt{-15}) \).
• 1.16.af : \(\Q(\sqrt{-39}) \).

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.ak_bd_abo	$2$	(not in LMFDB)
3.16.ag_ad_dk	$2$	(not in LMFDB)
3.16.ae_an_fw	$2$	(not in LMFDB)
3.16.e_an_afw	$2$	(not in LMFDB)
3.16.g_ad_adk	$2$	(not in LMFDB)
3.16.k_bd_bo	$2$	(not in LMFDB)
3.16.u_gx_bjk	$2$	(not in LMFDB)
3.16.ai_bj_aem	$3$	(not in LMFDB)