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

• Label: 3.16.at_gi_abga
• Base field: $\F_{2^{4}}$
• Dimension: $3$
• $p$-rank: $1$
• 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:	$( 1 - 4 x )^{2}( 1 - 7 x + 16 x^{2} )( 1 - 4 x + 16 x^{2} )$
	$1 - 19 x + 164 x^{2} - 832 x^{3} + 2624 x^{4} - 4864 x^{5} + 4096 x^{6}$
Frobenius angles:	$0$, $0$, $\pm0.160861246510$, $\pm0.333333333333$
Angle rank:	$1$ (numerical)

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1170$	$14742000$	$68585312250$	$281333762892000$	$1151383396862234250$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-2$	$224$	$4090$	$65504$	$1047178$	$16768976$	$268416538$	$4294983104$	$68719562410$	$1099509877424$

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 $\times$ 1.16.ah $\times$ 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.ai : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• 1.16.ah : \(\Q(\sqrt{-15}) \).
• 1.16.ae : \(\Q(\sqrt{-3}) \).

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

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

• 1.16777216.amdc^2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• 1.16777216.mbf : \(\Q(\sqrt{-15}) \).

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 $\times$ 1.256.ar $\times$ 1.256.q. The endomorphism algebra for each factor is:

  • 1.256.abg : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
  • 1.256.ar : \(\Q(\sqrt{-15}) \).
  • 1.256.q : \(\Q(\sqrt{-3}) \).
• Endomorphism algebra over $\F_{2^{12}}$
  The base change of $A$ to $\F_{2^{12}}$ is 1.4096.aey $\times$ 1.4096.ah $\times$ 1.4096.ey. The endomorphism algebra for each factor is:

  • 1.4096.aey : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
  • 1.4096.ah : \(\Q(\sqrt{-15}) \).
  • 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.al_bs_aey	$2$	(not in LMFDB)
3.16.af_ae_cm	$2$	(not in LMFDB)
3.16.ad_am_ey	$2$	(not in LMFDB)
3.16.d_am_aey	$2$	(not in LMFDB)
3.16.f_ae_acm	$2$	(not in LMFDB)
3.16.l_bs_ey	$2$	(not in LMFDB)
3.16.t_gi_bga	$2$	(not in LMFDB)
3.16.ah_aq_iq	$3$	(not in LMFDB)
3.16.ah_bg_aei	$3$	(not in LMFDB)
3.16.f_ae_acm	$3$	(not in LMFDB)