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

• Label: 6.2.ai_bd_acg_cf_i_adc
• Base field: $\F_{2}$
• Dimension: $6$
• $p$-rank: $4$
• 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:	$( 1 - 2 x + 2 x^{2} )^{2}( 1 - 4 x + 5 x^{2} + 2 x^{3} - 11 x^{4} + 4 x^{5} + 20 x^{6} - 32 x^{7} + 16 x^{8} )$
	$1 - 8 x + 29 x^{2} - 58 x^{3} + 57 x^{4} + 8 x^{5} - 80 x^{6} + 16 x^{7} + 228 x^{8} - 464 x^{9} + 464 x^{10} - 256 x^{11} + 64 x^{12}$
Frobenius angles:	$\pm0.0247483856139$, $\pm0.177336015878$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.344002682545$, $\pm0.858081718947$
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:	$4$
Slopes:	$[0, 0, 0, 0, 1/2, 1/2, 1/2, 1/2, 1, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1$	$1525$	$852436$	$48075625$	$1110571141$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-5$	$-1$	$19$	$35$	$35$	$65$	$65$	$227$	$415$	$959$

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 1.2.ac^2 $\times$ 4.2.ae_f_c_al 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 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
• 4.2.ae_f_c_al : 8.0.22581504.2.

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

The base change of $A$ to $\F_{2^{12}}$ is 1.4096.ey^2 $\times$ 2.4096.ahm_zkj^2. 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.ahm_zkj^2 : $\mathrm{M}_{2}($4.0.4752.1$)$

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 $\times$ 4.4.ag_t_abq_dd. The endomorphism algebra for each factor is:

  • 1.4.a^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
  • 4.4.ag_t_abq_dd : 8.0.22581504.2.
• Endomorphism algebra over $\F_{2^{3}}$
  The base change of $A$ to $\F_{2^{3}}$ is 1.8.e^2 $\times$ 4.8.c_c_ak_aep. The endomorphism algebra for each factor is:

  • 1.8.e^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
  • 4.8.c_c_ak_aep : 8.0.22581504.2.
• Endomorphism algebra over $\F_{2^{4}}$
  The base change of $A$ to $\F_{2^{4}}$ is 1.16.i^2 $\times$ 4.16.c_t_adq_adr. The endomorphism algebra for each factor is:

  • 1.16.i^2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
  • 4.16.c_t_adq_adr : 8.0.22581504.2.
• Endomorphism algebra over $\F_{2^{6}}$
  The base change of $A$ to $\F_{2^{6}}$ is 1.64.a^2 $\times$ 4.64.a_ahm_a_zkj. The endomorphism algebra for each factor is:

  • 1.64.a^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
  • 4.64.a_ahm_a_zkj : 8.0.22581504.2.

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
6.2.ae_f_c_ah_am_bo	$2$	(not in LMFDB)
6.2.a_ad_ac_j_a_aq	$2$	(not in LMFDB)
6.2.a_ad_c_j_a_aq	$2$	(not in LMFDB)
6.2.e_f_ac_ah_m_bo	$2$	(not in LMFDB)
6.2.i_bd_cg_cf_ai_adc	$2$	(not in LMFDB)
6.2.ac_ab_i_aj_ak_bi	$3$	(not in LMFDB)
6.2.ac_c_c_ad_i_ai	$3$	(not in LMFDB)
6.2.ac_f_ae_j_ae_q	$3$	(not in LMFDB)
6.2.e_i_o_v_ba_bi	$3$	(not in LMFDB)
6.2.e_l_ba_bz_di_fa	$3$	(not in LMFDB)