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

• Label: 4.2.ae_j_as_be
• Base field: $\F_{2}$
• Dimension: $4$
• $p$-rank: $2$
• 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:	$( 1 - 2 x + 2 x^{2} - 4 x^{3} + 4 x^{4} )( 1 - 2 x + 3 x^{2} - 4 x^{3} + 4 x^{4} )$
	$1 - 4 x + 9 x^{2} - 18 x^{3} + 30 x^{4} - 36 x^{5} + 36 x^{6} - 32 x^{7} + 16 x^{8}$
Frobenius angles:	$\pm0.0833333333333$, $\pm0.174442860055$, $\pm0.546783656212$, $\pm0.583333333333$
Angle rank:	$2$ (numerical)
Jacobians:	$0$
Isomorphism classes:	2

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2$	$364$	$1550$	$37856$	$2327602$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-1$	$7$	$-1$	$7$	$59$	$91$	$111$	$287$	$611$	$987$

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 $\F_{2}$

The isogeny class factors as 2.2.ac_c $\times$ 2.2.ac_d 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.ac_c : \(\Q(\zeta_{12})\).
• 2.2.ac_d : 4.0.1088.2.

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

The base change of $A$ to $\F_{2^{12}}$ is 1.4096.ey^2 $\times$ 2.4096.adu_hrl. The endomorphism algebra for each factor is:

• 1.4096.ey^2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• 2.4096.adu_hrl : 4.0.1088.2.

Remainder of endomorphism lattice by field

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

  • 2.4.a_ae : \(\Q(\zeta_{12})\).
  • 2.4.c_b : 4.0.1088.2.
• Endomorphism algebra over $\F_{2^{3}}$
  The base change of $A$ to $\F_{2^{3}}$ is 1.8.ae^2 $\times$ 2.8.ac_p. The endomorphism algebra for each factor is:

  • 1.8.ae^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
  • 2.8.ac_p : 4.0.1088.2.
• Endomorphism algebra over $\F_{2^{4}}$
  The base change of $A$ to $\F_{2^{4}}$ is 1.16.ae^2 $\times$ 2.16.ac_b. The endomorphism algebra for each factor is:

  • 1.16.ae^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
  • 2.16.ac_b : 4.0.1088.2.
• Endomorphism algebra over $\F_{2^{6}}$
  The base change of $A$ to $\F_{2^{6}}$ is 1.64.a^2 $\times$ 2.64.ba_ld. The endomorphism algebra for each factor is:

  • 1.64.a^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
  • 2.64.ba_ld : 4.0.1088.2.

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.a_b_ac_ac	$2$	4.4.c_ad_a_bc
4.2.a_b_c_ac	$2$	4.4.c_ad_a_bc
4.2.e_j_s_be	$2$	4.4.c_ad_a_bc
4.2.c_d_a_a	$3$	(not in LMFDB)