Abelian variety isogeny class 6.2.b_a_c_e_a_a over $\F_{2}$

• Label: 6.2.b_a_c_e_a_a
• Base field: $\F_{2}$
• Dimension: $6$
• $p$-rank: $1$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{2}$
Dimension:	$6$
L-polynomial:	$( 1 + x + 2 x^{3} + 4 x^{4} )( 1 + 16 x^{8} )$
	$1 + x + 2 x^{3} + 4 x^{4} + 16 x^{8} + 16 x^{9} + 32 x^{11} + 64 x^{12}$
Frobenius angles:	$\pm0.125000000000$, $\pm0.313829601922$, $\pm0.375000000000$, $\pm0.625000000000$, $\pm0.860613258134$, $\pm0.875000000000$
Angle rank:	$2$ (numerical)
Jacobians:	$0$

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$136$	$4624$	$622744$	$34744736$	$847250216$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$4$	$4$	$16$	$24$	$24$	$64$	$88$	$416$	$520$	$1104$

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 $\F_{2^{8}}$.

Endomorphism algebra over $\F_{2}$

The isogeny class factors as 2.2.b_a $\times$ 4.2.a_a_a_a 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.b_a : 4.0.2312.1.
• 4.2.a_a_a_a : \(\Q(\zeta_{16})\).

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

The base change of $A$ to $\F_{2^{8}}$ is 1.256.bg^4 $\times$ 2.256.bf_uy. The endomorphism algebra for each factor is:

• 1.256.bg^4 : $\mathrm{M}_{4}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• 2.256.bf_uy : 4.0.2312.1.

Remainder of endomorphism lattice by field

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

  • 2.4.ab_e : 4.0.2312.1.
  • 2.4.a_a^2 : $\mathrm{M}_{2}($\(\Q(\zeta_{8})\)$)$
• Endomorphism algebra over $\F_{2^{4}}$
  The base change of $A$ to $\F_{2^{4}}$ is 1.16.a^4 $\times$ 2.16.h_bo. The endomorphism algebra for each factor is:

  • 1.16.a^4 : $\mathrm{M}_{4}($\(\Q(\sqrt{-1}) \)$)$
  • 2.16.h_bo : 4.0.2312.1.

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
6.2.ab_a_ac_e_a_a	$2$	(not in LMFDB)
6.2.aj_bo_aek_jc_apc_we	$16$	(not in LMFDB)
6.2.ah_y_abu_bs_i_acm	$16$	(not in LMFDB)