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

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

Invariants

Base field:	$\F_{2^{2}}$
Dimension:	$4$
L-polynomial:	$( 1 + 2 x + 4 x^{2} )( 1 - 2 x - 3 x^{2} + 13 x^{3} - 12 x^{4} - 32 x^{5} + 64 x^{6} )$
	$1 - 3 x^{2} - x^{3} + 2 x^{4} - 4 x^{5} - 48 x^{6} + 256 x^{8}$
Frobenius angles:	$\pm0.0556608295517$, $\pm0.341375115266$, $\pm0.666666666667$, $\pm0.912803686695$
Angle rank:	$1$ (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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$203$	$43239$	$15399377$	$4155657051$	$1064508218753$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$5$	$11$	$62$	$247$	$990$	$3788$	$16252$	$66343$	$261323$	$1052396$

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 $\F_{2^{42}}$.

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

The isogeny class factors as 1.4.c $\times$ 3.4.ac_ad_n 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.c : \(\Q(\sqrt{-3}) \).
• 3.4.ac_ad_n : \(\Q(\zeta_{7})\).

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

The base change of $A$ to $\F_{2^{42}}$ is 1.4398046511104.ajeqpk $\times$ 1.4398046511104.hxvrd^3. The endomorphism algebra for each factor is:

• 1.4398046511104.ajeqpk : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• 1.4398046511104.hxvrd^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-7}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{2^{6}}$
  The base change of $A$ to $\F_{2^{6}}$ is 1.64.aq $\times$ 3.64.n_ag_abcp. The endomorphism algebra for each factor is:

  • 1.64.aq : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
  • 3.64.n_ag_abcp : \(\Q(\zeta_{7})\).
• Endomorphism algebra over $\F_{2^{14}}$
  The base change of $A$ to $\F_{2^{14}}$ is 1.16384.adj^3 $\times$ 1.16384.ey. The endomorphism algebra for each factor is:

  • 1.16384.adj^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-7}) \)$)$
  • 1.16384.ey : \(\Q(\sqrt{-3}) \).

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_f_l_aby	$2$	(not in LMFDB)
4.4.a_ad_b_c	$2$	(not in LMFDB)
4.4.e_f_al_aby	$2$	(not in LMFDB)
4.4.ag_j_r_acy	$3$	(not in LMFDB)