Abelian variety isogeny class 4.4.ak_bw_aft_nc over $\F_{2^{2}}$

• Label: 4.4.ak_bw_aft_nc
• Base field: $\F_{2^{2}}$
• Dimension: $4$
• $p$-rank: $3$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{2^{2}}$
Dimension:	$4$
L-polynomial:	$( 1 - 2 x )^{2}( 1 - 3 x + 4 x^{2} )( 1 - 3 x + 7 x^{2} - 12 x^{3} + 16 x^{4} )$
	$1 - 10 x + 48 x^{2} - 149 x^{3} + 340 x^{4} - 596 x^{5} + 768 x^{6} - 640 x^{7} + 256 x^{8}$
Frobenius angles:	$0$, $0$, $\pm0.190783854037$, $\pm0.230053456163$, $\pm0.524117187371$
Angle rank:	$2$ (numerical)

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

Newton polygon

$p$-rank:	$3$
Slopes:	$[0, 0, 0, 1/2, 1/2, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$18$	$50544$	$15272712$	$4116808800$	$1186724982798$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-5$	$13$	$58$	$249$	$1105$	$4246$	$16039$	$64113$	$260122$	$1044853$

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 $\F_{2^{2}}$

The isogeny class factors as 1.4.ae $\times$ 1.4.ad $\times$ 2.4.ad_h and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.4.ae : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• 1.4.ad : \(\Q(\sqrt{-7}) \).
• 2.4.ad_h : \(\Q(\sqrt{-3}, \sqrt{13})\).

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

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

• 1.4096.aey : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• 1.4096.bv : \(\Q(\sqrt{-7}) \).
• 1.4096.el^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-39}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{2^{4}}$
  The base change of $A$ to $\F_{2^{4}}$ is 1.16.ai $\times$ 1.16.ab $\times$ 2.16.f_j. The endomorphism algebra for each factor is:

  • 1.16.ai : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
  • 1.16.ab : \(\Q(\sqrt{-7}) \).
  • 2.16.f_j : \(\Q(\sqrt{-3}, \sqrt{13})\).
• Endomorphism algebra over $\F_{2^{6}}$
  The base change of $A$ to $\F_{2^{6}}$ is 1.64.aq $\times$ 1.64.j $\times$ 2.64.a_el. The endomorphism algebra for each factor is:

  • 1.64.aq : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
  • 1.64.j : \(\Q(\sqrt{-7}) \).
  • 2.64.a_el : \(\Q(\sqrt{-3}, \sqrt{13})\).

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
4.4.ae_g_al_bc	$2$	(not in LMFDB)
4.4.ae_g_af_e	$2$	(not in LMFDB)
4.4.ac_a_l_au	$2$	(not in LMFDB)
4.4.c_a_al_au	$2$	(not in LMFDB)
4.4.e_g_f_e	$2$	(not in LMFDB)
4.4.e_g_l_bc	$2$	(not in LMFDB)
4.4.k_bw_ft_nc	$2$	(not in LMFDB)
4.4.ah_p_h_acq	$3$	(not in LMFDB)
4.4.ae_g_af_e	$3$	(not in LMFDB)
4.4.ae_m_abd_cs	$3$	(not in LMFDB)
4.4.ab_ad_b_w	$3$	(not in LMFDB)
4.4.c_g_h_w	$3$	(not in LMFDB)