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

• Label: 6.2.ai_bc_abx_t_ec_ajo
• 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:	$( 1 - 2 x + 2 x^{2} )^{2}( 1 - 4 x + 4 x^{2} + 7 x^{3} - 21 x^{4} + 14 x^{5} + 16 x^{6} - 32 x^{7} + 16 x^{8} )$
	$1 - 8 x + 28 x^{2} - 49 x^{3} + 19 x^{4} + 106 x^{5} - 248 x^{6} + 212 x^{7} + 76 x^{8} - 392 x^{9} + 448 x^{10} - 256 x^{11} + 64 x^{12}$
Frobenius angles:	$\pm0.0764513550391$, $\pm0.143118021706$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.323548644961$, $\pm0.943118021706$
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:	$4$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1$	$775$	$1262599$	$34894375$	$2258055361$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-5$	$-3$	$22$	$29$	$55$	$54$	$114$	$237$	$544$	$1147$

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 $\F_{2^{60}}$.

Endomorphism algebra over $\F_{2}$

The isogeny class factors as 1.2.ac^2 $\times$ 4.2.ae_e_h_av 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_e_h_av : \(\Q(\zeta_{15})\).

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

The base change of $A$ to $\F_{2^{60}}$ is 1.1152921504606846976.bwvqqfv^4 $\times$ 1.1152921504606846976.gytisyy^2. The endomorphism algebra for each factor is:

• 1.1152921504606846976.bwvqqfv^4 : $\mathrm{M}_{4}($\(\Q(\sqrt{-15}) \)$)$
• 1.1152921504606846976.gytisyy^2 : $\mathrm{M}_{2}(B)$, 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^2 $\times$ 4.4.ai_be_acv_ft. The endomorphism algebra for each factor is:

  • 1.4.a^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
  • 4.4.ai_be_acv_ft : \(\Q(\zeta_{15})\).
• Endomorphism algebra over $\F_{2^{3}}$
  The base change of $A$ to $\F_{2^{3}}$ is 1.8.e^2 $\times$ 4.8.f_h_z_ez. The endomorphism algebra for each factor is:

  • 1.8.e^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
  • 4.8.f_h_z_ez : \(\Q(\zeta_{15})\).
• Endomorphism algebra over $\F_{2^{4}}$
  The base change of $A$ to $\F_{2^{4}}$ is 1.16.i^2 $\times$ 4.16.ae_be_adx_ban. 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.ae_be_adx_ban : \(\Q(\zeta_{15})\).
• Endomorphism algebra over $\F_{2^{5}}$
  The base change of $A$ to $\F_{2^{5}}$ is 1.32.i^2 $\times$ 2.32.d_bj^2. The endomorphism algebra for each factor is:

  • 1.32.i^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
  • 2.32.d_bj^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}, \sqrt{5})\)$)$
• Endomorphism algebra over $\F_{2^{6}}$
  The base change of $A$ to $\F_{2^{6}}$ is 1.64.a^2 $\times$ 4.64.al_cf_cz_agqx. The endomorphism algebra for each factor is:

  • 1.64.a^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
  • 4.64.al_cf_cz_agqx : \(\Q(\zeta_{15})\).
• Endomorphism algebra over $\F_{2^{10}}$
  The base change of $A$ to $\F_{2^{10}}$ is 1.1024.a^2 $\times$ 2.1024.cj_dzt^2. The endomorphism algebra for each factor is:

  • 1.1024.a^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
  • 2.1024.cj_dzt^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}, \sqrt{5})\)$)$
• Endomorphism algebra over $\F_{2^{12}}$
  The base change of $A$ to $\F_{2^{12}}$ is 1.4096.ey^2 $\times$ 4.4096.ah_afzr_dgij_bjklft. 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 $\infty$.
  • 4.4096.ah_afzr_dgij_bjklft : \(\Q(\zeta_{15})\).
• Endomorphism algebra over $\F_{2^{15}}$
  The base change of $A$ to $\F_{2^{15}}$ is 1.32768.ajw^2 $\times$ 2.32768.a_acgor^2. The endomorphism algebra for each factor is:

  • 1.32768.ajw^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
  • 2.32768.a_acgor^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}, \sqrt{5})\)$)$
• Endomorphism algebra over $\F_{2^{20}}$
  The base change of $A$ to $\F_{2^{20}}$ is 1.1048576.dau^2 $\times$ 2.1048576.cmj_dvphh^2. The endomorphism algebra for each factor is:

  • 1.1048576.dau^2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
  • 2.1048576.cmj_dvphh^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}, \sqrt{5})\)$)$
• Endomorphism algebra over $\F_{2^{30}}$
  The base change of $A$ to $\F_{2^{30}}$ is 1.1073741824.acgor^4 $\times$ 1.1073741824.a^2. The endomorphism algebra for each factor is:

  • 1.1073741824.acgor^4 : $\mathrm{M}_{4}($\(\Q(\sqrt{-15}) \)$)$
  • 1.1073741824.a^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$

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.ae_e_h_ar_ac_bg	$2$	(not in LMFDB)
6.2.a_ae_ab_l_c_ay	$2$	(not in LMFDB)
6.2.a_ae_b_l_ac_ay	$2$	(not in LMFDB)
6.2.e_e_ah_ar_c_bg	$2$	(not in LMFDB)
6.2.i_bc_bx_t_aec_ajo	$2$	(not in LMFDB)
6.2.ac_ac_l_al_ao_bu	$3$	(not in LMFDB)
6.2.b_b_f_h_k_q	$3$	(not in LMFDB)
6.2.b_e_i_n_w_bi	$3$	(not in LMFDB)
6.2.h_z_cn_fj_jq_os	$3$	(not in LMFDB)