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

• Label: 3.16.as_fx_abds
• 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:	$( 1 - 4 x + 16 x^{2} )( 1 - 7 x + 16 x^{2} )^{2}$
	$1 - 18 x + 153 x^{2} - 772 x^{3} + 2448 x^{4} - 4608 x^{5} + 4096 x^{6}$
Frobenius angles:	$\pm0.160861246510$, $\pm0.160861246510$, $\pm0.333333333333$
Angle rank:	$1$ (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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1300$	$15724800$	$70676222500$	$284513775436800$	$1155477213297062500$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-1$	$239$	$4211$	$66239$	$1050899$	$16785311$	$268479539$	$4295195519$	$68720172371$	$1099511272799$

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 $\F_{2^{12}}$.

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

The isogeny class factors as 1.16.ah^2 $\times$ 1.16.ae 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.ah^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$
• 1.16.ae : \(\Q(\sqrt{-3}) \).

Endomorphism algebra over $\overline{\F}_{2^{4}}$

The base change of $A$ to $\F_{2^{12}}$ is 1.4096.ah^2 $\times$ 1.4096.ey. The endomorphism algebra for each factor is:

• 1.4096.ah^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$
• 1.4096.ey : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.

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_bp_aeu	$2$	(not in LMFDB)
3.16.ae_ab_cq	$2$	(not in LMFDB)
3.16.e_ab_acq	$2$	(not in LMFDB)
3.16.k_bp_eu	$2$	(not in LMFDB)
3.16.s_fx_bds	$2$	(not in LMFDB)
3.16.ag_ap_hs	$3$	(not in LMFDB)
3.16.d_v_do	$3$	(not in LMFDB)
3.16.p_eb_su	$3$	(not in LMFDB)