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

• Label: 4.2.ad_e_ad_d
• Base field: $\F_{2}$
• Dimension: $4$
• $p$-rank: $4$
• Ordinary: yes
• 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:	$( 1 - 3 x + 5 x^{2} - 6 x^{3} + 4 x^{4} )( 1 - x^{2} + 4 x^{4} )$
	$1 - 3 x + 4 x^{2} - 3 x^{3} + 3 x^{4} - 6 x^{5} + 16 x^{6} - 24 x^{7} + 16 x^{8}$
Frobenius angles:	$\pm0.123548644961$, $\pm0.209784688372$, $\pm0.456881978294$, $\pm0.790215311628$
Angle rank:	$1$ (numerical)
Jacobians:	$0$
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$4$
Slopes:	$[0, 0, 0, 0, 1, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4$	$304$	$5776$	$98496$	$926404$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$0$	$4$	$9$	$24$	$30$	$109$	$168$	$240$	$513$	$964$

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 $\F_{2^{6}}$.

Endomorphism algebra over $\F_{2}$

The isogeny class factors as 2.2.ad_f $\times$ 2.2.a_ab and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 2.2.ad_f : \(\Q(\sqrt{-3}, \sqrt{5})\).
• 2.2.a_ab : \(\Q(\sqrt{-3}, \sqrt{5})\).

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

The base change of $A$ to $\F_{2^{6}}$ is 1.64.l^4 and its endomorphism algebra is $\mathrm{M}_{4}($\(\Q(\sqrt{-15}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{2^{2}}$
  The base change of $A$ to $\F_{2^{2}}$ is 1.4.ab^2 $\times$ 2.4.b_ad. The endomorphism algebra for each factor is:

  • 1.4.ab^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$
  • 2.4.b_ad : \(\Q(\sqrt{-3}, \sqrt{5})\).
• Endomorphism algebra over $\F_{2^{3}}$

  The base change of $A$ to $\F_{2^{3}}$ is 2.8.a_l^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}, \sqrt{5})\)$)$

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.d_e_d_d	$2$	4.4.ab_e_l_al
4.2.ag_t_abq_cr	$3$	(not in LMFDB)
4.2.a_ac_a_j	$3$	(not in LMFDB)
4.2.a_b_a_ad	$3$	(not in LMFDB)
4.2.d_e_d_d	$3$	(not in LMFDB)
4.2.g_t_bq_cr	$3$	(not in LMFDB)