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

• Label: 3.16.ar_fb_ayi
• 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:	$( 1 - 4 x )^{2}( 1 - 9 x + 43 x^{2} - 144 x^{3} + 256 x^{4} )$
	$1 - 17 x + 131 x^{2} - 632 x^{3} + 2096 x^{4} - 4352 x^{5} + 4096 x^{6}$
Frobenius angles:	$0$, $0$, $\pm0.108303609292$, $\pm0.441636942625$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1323$	$14982975$	$66603519108$	$277216037275275$	$1149181225223419773$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$0$	$230$	$3969$	$64538$	$1045170$	$16776455$	$268457868$	$4294942322$	$68718952449$	$1099511012150$

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 $\F_{2^{24}}$.

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

The isogeny class factors as 1.16.ai $\times$ 2.16.aj_br 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_br : \(\Q(\sqrt{-3}, \sqrt{37})\).

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

The base change of $A$ to $\F_{2^{24}}$ is 1.16777216.amdc $\times$ 1.16777216.fmx^2. The endomorphism algebra for each factor is:

• 1.16777216.amdc : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• 1.16777216.fmx^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-111}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{2^{8}}$
  The base change of $A$ to $\F_{2^{8}}$ is 1.256.abg $\times$ 2.256.f_aix. The endomorphism algebra for each factor is:

  • 1.256.abg : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
  • 2.256.f_aix : \(\Q(\sqrt{-3}, \sqrt{37})\).
• Endomorphism algebra over $\F_{2^{12}}$
  The base change of $A$ to $\F_{2^{12}}$ is 1.4096.aey $\times$ 2.4096.a_fmx. The endomorphism algebra for each factor is:

  • 1.4096.aey : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
  • 2.4096.a_fmx : \(\Q(\sqrt{-3}, \sqrt{37})\).

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_an_ce	$2$	(not in LMFDB)
3.16.b_an_ace	$2$	(not in LMFDB)
3.16.r_fb_yi	$2$	(not in LMFDB)
3.16.ai_l_bo	$3$	(not in LMFDB)
3.16.af_x_aem	$3$	(not in LMFDB)
3.16.e_l_au	$3$	(not in LMFDB)
3.16.n_dr_rs	$3$	(not in LMFDB)