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

• Label: 4.2.af_q_abi_ce
• Base field: $\F_{2}$
• Dimension: $4$
• $p$-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:	$4$
L-polynomial:	$( 1 - x + 2 x^{2} )( 1 + 2 x^{2} )( 1 - 2 x + 2 x^{2} )^{2}$
	$1 - 5 x + 16 x^{2} - 34 x^{3} + 56 x^{4} - 68 x^{5} + 64 x^{6} - 40 x^{7} + 16 x^{8}$
Frobenius angles:	$\pm0.250000000000$, $\pm0.250000000000$, $\pm0.384973271919$, $\pm0.5$
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:	$1$
Slopes:	$[0, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6$	$1800$	$21294$	$90000$	$1220406$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-2$	$12$	$22$	$24$	$38$	$72$	$110$	$192$	$454$	$1032$

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 1.2.ac^2 $\times$ 1.2.ab $\times$ 1.2.a 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^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
• 1.2.ab : \(\Q(\sqrt{-7}) \).
• 1.2.a : \(\Q(\sqrt{-2}) \).

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

The base change of $A$ to $\F_{2^{8}}$ is 1.256.abg^3 $\times$ 1.256.bf. The endomorphism algebra for each factor is:

• 1.256.abg^3 : $\mathrm{M}_{3}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• 1.256.bf : \(\Q(\sqrt{-7}) \).

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^2 $\times$ 1.4.d $\times$ 1.4.e. The endomorphism algebra for each factor is:

  • 1.4.a^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
  • 1.4.d : \(\Q(\sqrt{-7}) \).
  • 1.4.e : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• Endomorphism algebra over $\F_{2^{4}}$
  The base change of $A$ to $\F_{2^{4}}$ is 1.16.ai $\times$ 1.16.ab $\times$ 1.16.i^2. The endomorphism algebra for each factor is:

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

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_i_ao_y	$2$	4.4.h_bc_dg_hk
4.2.ab_e_ac_i	$2$	4.4.h_bc_dg_hk
4.2.b_e_c_i	$2$	4.4.h_bc_dg_hk
4.2.d_i_o_y	$2$	4.4.h_bc_dg_hk
4.2.f_q_bi_ce	$2$	4.4.h_bc_dg_hk
4.2.b_e_i_i	$3$	(not in LMFDB)