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

• Label: 6.2.ak_by_agi_pg_abcy_bsy
• Base field: $\F_{2}$
• Dimension: $6$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{2}$
Dimension:	$6$
L-polynomial:	$( 1 - 2 x + 2 x^{2} )^{4}( 1 - 2 x + 2 x^{2} - 4 x^{3} + 4 x^{4} )$
	$1 - 10 x + 50 x^{2} - 164 x^{3} + 396 x^{4} - 752 x^{5} + 1168 x^{6} - 1504 x^{7} + 1584 x^{8} - 1312 x^{9} + 800 x^{10} - 320 x^{11} + 64 x^{12}$
Frobenius angles:	$\pm0.0833333333333$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.583333333333$
Angle rank:	$0$ (numerical)
Jacobians:	$0$

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

Newton polygon

This isogeny class is supersingular.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1$	$8125$	$714025$	$66015625$	$3732830281$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-7$	$5$	$17$	$41$	$73$	$65$	$49$	$161$	$449$	$1025$

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^4 $\times$ 2.2.ac_c 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^4 : $\mathrm{M}_{4}($\(\Q(\sqrt{-1}) \)$)$
• 2.2.ac_c : \(\Q(\zeta_{12})\).

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

The base change of $A$ to $\F_{2^{12}}$ is 1.4096.ey^6 and its endomorphism algebra is $\mathrm{M}_{6}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.

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^4 $\times$ 2.4.a_ae. The endomorphism algebra for each factor is:

  • 1.4.a^4 : $\mathrm{M}_{4}($\(\Q(\sqrt{-1}) \)$)$
  • 2.4.a_ae : \(\Q(\zeta_{12})\).
• Endomorphism algebra over $\F_{2^{3}}$
  The base change of $A$ to $\F_{2^{3}}$ is 1.8.ae^2 $\times$ 1.8.e^4. The endomorphism algebra for each factor is:

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

  • 1.16.ae^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
  • 1.16.i^4 : $\mathrm{M}_{4}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• Endomorphism algebra over $\F_{2^{6}}$

  The base change of $A$ to $\F_{2^{6}}$ is 1.64.a^6 and its endomorphism algebra is $\mathrm{M}_{6}($\(\Q(\sqrt{-1}) \)$)$

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.ag_s_abk_ci_ads_fo	$2$	(not in LMFDB)
6.2.ag_s_abc_m_bw_aei	$2$	(not in LMFDB)
6.2.ac_c_ae_m_aq_q	$2$	(not in LMFDB)
6.2.ac_c_e_ae_a_q	$2$	(not in LMFDB)
6.2.c_c_ae_ae_a_q	$2$	(not in LMFDB)
6.2.c_c_e_m_q_q	$2$	(not in LMFDB)
6.2.g_s_bc_m_abw_aei	$2$	(not in LMFDB)
6.2.g_s_bk_ci_ds_fo	$2$	(not in LMFDB)
6.2.k_by_gi_pg_bcy_bsy	$2$	(not in LMFDB)
6.2.ae_i_ai_a_q_abg	$3$	(not in LMFDB)
6.2.ae_i_ai_m_abg_cm	$3$	(not in LMFDB)
6.2.c_c_e_a_ai_ai	$3$	(not in LMFDB)
6.2.c_c_e_m_q_q	$3$	(not in LMFDB)
6.2.i_bg_dk_hk_no_uy	$3$	(not in LMFDB)