Abelian variety isogeny class 4.2.af_r_abl_ck over $\F_{2}$

• Label: 4.2.af_r_abl_ck
• Base field: $\F_{2}$
• 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:	$\F_{2}$
Dimension:	$4$
L-polynomial:	$( 1 - 2 x + 2 x^{2} )( 1 - x + 2 x^{2} )^{3}$
	$1 - 5 x + 17 x^{2} - 37 x^{3} + 62 x^{4} - 74 x^{5} + 68 x^{6} - 40 x^{7} + 16 x^{8}$
Frobenius angles:	$\pm0.250000000000$, $\pm0.384973271919$, $\pm0.384973271919$, $\pm0.384973271919$
Angle rank:	$1$ (numerical)
Jacobians:	$0$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$8$	$2560$	$35672$	$102400$	$436568$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-2$	$14$	$28$	$22$	$8$	$38$	$152$	$318$	$496$	$854$

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 $\F_{2^{4}}$.

Endomorphism algebra over $\F_{2}$

The isogeny class factors as 1.2.ac $\times$ 1.2.ab^3 and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.2.ac : \(\Q(\sqrt{-1}) \).
• 1.2.ab^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-7}) \)$)$

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

The base change of $A$ to $\F_{2^{4}}$ is 1.16.ab^3 $\times$ 1.16.i. The endomorphism algebra for each factor is:

• 1.16.ab^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-7}) \)$)$
• 1.16.i : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{2^{2}}$
  The base change of $A$ to $\F_{2^{2}}$ is 1.4.a $\times$ 1.4.d^3. The endomorphism algebra for each factor is:

  • 1.4.a : \(\Q(\sqrt{-1}) \).
  • 1.4.d^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-7}) \)$)$

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
4.2.ad_j_ap_ba	$2$	4.4.j_br_ff_ma
4.2.ab_f_af_o	$2$	4.4.j_br_ff_ma
4.2.ab_f_ab_k	$2$	4.4.j_br_ff_ma
4.2.b_f_b_k	$2$	4.4.j_br_ff_ma
4.2.b_f_f_o	$2$	4.4.j_br_ff_ma
4.2.d_j_p_ba	$2$	4.4.j_br_ff_ma
4.2.f_r_bl_ck	$2$	4.4.j_br_ff_ma
4.2.ac_c_f_ak	$3$	(not in LMFDB)