Abelian variety isogeny class 3.8.aj_bt_afs over $\F_{2^{3}}$

• Label: 3.8.aj_bt_afs
• Base field: $\F_{2^{3}}$
• Dimension: $3$
• $p$-rank: $2$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes

Invariants

Base field:	$\F_{2^{3}}$
Dimension:	$3$
L-polynomial:	$( 1 - 4 x + 8 x^{2} )( 1 - 5 x + 17 x^{2} - 40 x^{3} + 64 x^{4} )$
	$1 - 9 x + 45 x^{2} - 148 x^{3} + 360 x^{4} - 576 x^{5} + 512 x^{6}$
Frobenius angles:	$\pm0.178413517577$, $\pm0.250000000000$, $\pm0.488253149089$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$185$	$305435$	$146236580$	$70062207475$	$35759343084925$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$0$	$74$	$555$	$4178$	$33300$	$264143$	$2097900$	$16763042$	$134186235$	$1073751914$

Jacobians and polarizations

This isogeny class is principally polarizable, but it is unknown whether it contains a Jacobian.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{36}}$.

Endomorphism algebra over $\F_{2^{3}}$

The isogeny class factors as 1.8.ae $\times$ 2.8.af_r and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.8.ae : \(\Q(\sqrt{-1}) \).
• 2.8.af_r : \(\Q(\sqrt{-3}, \sqrt{-7})\).

Endomorphism algebra over $\overline{\F}_{2^{3}}$

The base change of $A$ to $\F_{2^{36}}$ is 1.68719476736.abaytt^2 $\times$ 1.68719476736.bdvoy. The endomorphism algebra for each factor is:

• 1.68719476736.abaytt^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$
• 1.68719476736.bdvoy : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{2^{6}}$
  The base change of $A$ to $\F_{2^{6}}$ is 1.64.a $\times$ 2.64.j_r. The endomorphism algebra for each factor is:

  • 1.64.a : \(\Q(\sqrt{-1}) \).
  • 2.64.j_r : \(\Q(\sqrt{-3}, \sqrt{-7})\).
• Endomorphism algebra over $\F_{2^{9}}$
  The base change of $A$ to $\F_{2^{9}}$ is 1.512.f^2 $\times$ 1.512.bg. The endomorphism algebra for each factor is:

  • 1.512.f^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$
  • 1.512.bg : \(\Q(\sqrt{-1}) \).
• Endomorphism algebra over $\F_{2^{12}}$
  The base change of $A$ to $\F_{2^{12}}$ is 1.4096.ey $\times$ 2.4096.abv_acup. The endomorphism algebra for each factor is:

  • 1.4096.ey : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
  • 2.4096.abv_acup : \(\Q(\sqrt{-3}, \sqrt{-7})\).
• Endomorphism algebra over $\F_{2^{18}}$
  The base change of $A$ to $\F_{2^{18}}$ is 1.262144.a $\times$ 1.262144.bml^2. The endomorphism algebra for each factor is:

  • 1.262144.a : \(\Q(\sqrt{-1}) \).
  • 1.262144.bml^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$

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
3.8.ab_f_am	$2$	(not in LMFDB)
3.8.b_f_m	$2$	(not in LMFDB)
3.8.j_bt_fs	$2$	(not in LMFDB)
3.8.g_j_ae	$3$	(not in LMFDB)