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

• Label: 4.4.al_cg_aho_rw
• Base field: $\F_{2^{2}}$
• 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:	$\F_{2^{2}}$
Dimension:	$4$
L-polynomial:	$( 1 - 2 x )^{4}( 1 - 2 x + 4 x^{2} )( 1 - x + 4 x^{2} )$
	$1 - 11 x + 58 x^{2} - 196 x^{3} + 464 x^{4} - 784 x^{5} + 928 x^{6} - 704 x^{7} + 256 x^{8}$
Frobenius angles:	$0$, $0$, $0$, $0$, $\pm0.333333333333$, $\pm0.419569376745$
Angle rank:	$1$ (numerical)

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$12$	$40824$	$14780556$	$3316950000$	$884042324292$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-6$	$12$	$60$	$192$	$804$	$3720$	$15996$	$64992$	$260340$	$1043832$

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 $\F_{2^{2}}$

The isogeny class factors as 1.4.ae^2 $\times$ 1.4.ac $\times$ 1.4.ab 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.ae^2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• 1.4.ac : \(\Q(\sqrt{-3}) \).
• 1.4.ab : \(\Q(\sqrt{-15}) \).

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

The base change of $A$ to $\F_{2^{12}}$ is 1.4096.aey^3 $\times$ 1.4096.h. The endomorphism algebra for each factor is:

• 1.4096.aey^3 : $\mathrm{M}_{3}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• 1.4096.h : \(\Q(\sqrt{-15}) \).

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{2^{4}}$
  The base change of $A$ to $\F_{2^{4}}$ is 1.16.ai^2 $\times$ 1.16.e $\times$ 1.16.h. The endomorphism algebra for each factor is:

  • 1.16.ai^2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
  • 1.16.e : \(\Q(\sqrt{-3}) \).
  • 1.16.h : \(\Q(\sqrt{-15}) \).
• Endomorphism algebra over $\F_{2^{6}}$
  The base change of $A$ to $\F_{2^{6}}$ is 1.64.aq^2 $\times$ 1.64.l $\times$ 1.64.q. The endomorphism algebra for each factor is:

  • 1.64.aq^2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
  • 1.64.l : \(\Q(\sqrt{-15}) \).
  • 1.64.q : 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
4.4.aj_bm_aee_jg	$2$	(not in LMFDB)
4.4.ah_w_aca_ei	$2$	(not in LMFDB)
4.4.af_k_abc_dc	$2$	(not in LMFDB)
4.4.ad_c_m_abw	$2$	(not in LMFDB)
4.4.ab_ac_e_aq	$2$	(not in LMFDB)
4.4.b_ac_ae_aq	$2$	(not in LMFDB)
4.4.d_c_am_abw	$2$	(not in LMFDB)
4.4.f_k_bc_dc	$2$	(not in LMFDB)
4.4.h_w_ca_ei	$2$	(not in LMFDB)
4.4.j_bm_ee_jg	$2$	(not in LMFDB)
4.4.l_cg_ho_rw	$2$	(not in LMFDB)
4.4.af_e_u_acm	$3$	(not in LMFDB)
4.4.af_q_abo_dc	$3$	(not in LMFDB)
4.4.b_ac_ae_aq	$3$	(not in LMFDB)
4.4.b_k_i_ce	$3$	(not in LMFDB)
4.4.h_bc_dc_gu	$3$	(not in LMFDB)