Properties

Label 6.2.ai_be_acq_dy_aei_ey
Base Field $\F_{2}$
Dimension $6$
Ordinary No
$p$-rank $0$
Contains a Jacobian No

Learn more about

Invariants

Base field:  $\F_{2}$
Dimension:  $6$
L-polynomial:  $( 1 - 2 x + 2 x^{2} )^{2}( 1 - 4 x + 6 x^{2} - 4 x^{3} + 2 x^{4} - 8 x^{5} + 24 x^{6} - 32 x^{7} + 16 x^{8} )$
Frobenius angles:  $\pm0.0377785699724$, $\pm0.148391828106$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.398391828106$, $\pm0.787778569972$
Angle rank:  $2$ (numerical)

This isogeny class is not simple.

Newton polygon

$p$-rank:  $0$
Slopes:  $[1/4, 1/4, 1/4, 1/4, 1/2, 1/2, 1/2, 1/2, 3/4, 3/4, 3/4, 3/4]$

Point counts

This isogeny class does not contain a Jacobian, and it is unknown whether it is principally polarizable.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 1 2425 454441 42255625 769227281 80447418025 4133278016393 231405704600625 16960909411229953 1003619006560560625

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -5 1 13 33 25 73 121 209 481 881

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
The isogeny class factors as 1.2.ac 2 $\times$ 4.2.ae_g_ae_c and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{2}$
The base change of $A$ to $\F_{2^{8}}$ is 1.256.abg 2 $\times$ 4.256.q_ds_glk_jitk. The endomorphism algebra for each factor is:
  • 1.256.abg 2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
  • 4.256.q_ds_glk_jitk : the quaternion algebra over 4.0.1088.2 with the following ramification data at primes above $2$, and unramified at all archimedean places:
$v$ ($ 2 $,\( \pi \)) ($ 2 $,\( \pi + 1 \))
$\operatorname{inv}_v$$1/2$$1/2$
where $\pi$ is a root of $x^{4} - 2x^{3} + 5x^{2} - 4x + 2$.\n
All geometric endomorphisms are defined over $\F_{2^{8}}$.
Remainder of endomorphism lattice by field

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
6.2.ae_g_ae_g_ay_bw$2$(not in LMFDB)
6.2.a_ac_ae_g_a_a$2$(not in LMFDB)
6.2.a_ac_e_g_a_a$2$(not in LMFDB)
6.2.e_g_e_g_y_bw$2$(not in LMFDB)
6.2.i_be_cq_dy_ei_ey$2$(not in LMFDB)
6.2.ac_a_e_ag_ae_u$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
6.2.ae_g_ae_g_ay_bw$2$(not in LMFDB)
6.2.a_ac_ae_g_a_a$2$(not in LMFDB)
6.2.a_ac_e_g_a_a$2$(not in LMFDB)
6.2.e_g_e_g_y_bw$2$(not in LMFDB)
6.2.i_be_cq_dy_ei_ey$2$(not in LMFDB)
6.2.ac_a_e_ag_ae_u$3$(not in LMFDB)
6.2.ae_k_au_bm_acm_ds$4$(not in LMFDB)
6.2.ae_k_am_g_q_abg$4$(not in LMFDB)
6.2.a_c_ae_g_ai_q$4$(not in LMFDB)
6.2.a_c_e_g_i_q$4$(not in LMFDB)
6.2.e_k_m_g_aq_abg$4$(not in LMFDB)
6.2.e_k_u_bm_cm_ds$4$(not in LMFDB)
6.2.ag_q_abc_bq_aci_dg$6$(not in LMFDB)
6.2.c_a_ae_ag_e_u$6$(not in LMFDB)
6.2.g_q_bc_bq_ci_dg$6$(not in LMFDB)
6.2.ag_s_abk_cc_acq_dk$8$(not in LMFDB)
6.2.ae_c_m_as_ai_bo$8$(not in LMFDB)
6.2.ae_k_au_be_abo_ce$8$(not in LMFDB)
6.2.ac_c_ae_g_am_y$8$(not in LMFDB)
6.2.ac_g_am_w_abk_ce$8$(not in LMFDB)
6.2.ac_g_ae_g_m_ai$8$(not in LMFDB)
6.2.a_ac_ae_ac_i_i$8$(not in LMFDB)
6.2.a_ac_e_ac_ai_i$8$(not in LMFDB)
6.2.a_g_ae_o_ay_y$8$(not in LMFDB)
6.2.a_g_e_o_y_y$8$(not in LMFDB)
6.2.c_c_e_g_m_y$8$(not in LMFDB)
6.2.c_g_e_g_am_ai$8$(not in LMFDB)
6.2.c_g_m_w_bk_ce$8$(not in LMFDB)
6.2.e_c_am_as_i_bo$8$(not in LMFDB)
6.2.e_k_u_be_bo_ce$8$(not in LMFDB)
6.2.g_s_bk_cc_cq_dk$8$(not in LMFDB)
6.2.ac_e_am_s_abc_ca$12$(not in LMFDB)
6.2.ac_e_ae_c_e_am$12$(not in LMFDB)
6.2.c_e_e_c_ae_am$12$(not in LMFDB)
6.2.c_e_m_s_bc_ca$12$(not in LMFDB)
6.2.ae_e_i_as_ai_bw$16$(not in LMFDB)
6.2.ae_m_ay_bu_acu_ei$16$(not in LMFDB)
6.2.ac_a_e_ac_ae_i$16$(not in LMFDB)
6.2.ac_i_am_be_abk_cu$16$(not in LMFDB)
6.2.a_ai_a_be_a_acu$16$(not in LMFDB)
6.2.a_ae_a_o_a_abg$16$(not in LMFDB)
6.2.a_a_a_ac_a_ai$16$(not in LMFDB)
6.2.a_a_a_ac_a_i$16$(not in LMFDB)
6.2.a_e_a_o_a_bg$16$(not in LMFDB)
6.2.a_i_a_be_a_cu$16$(not in LMFDB)
6.2.c_a_ae_ac_e_i$16$(not in LMFDB)
6.2.c_i_m_be_bk_cu$16$(not in LMFDB)
6.2.e_e_ai_as_i_bw$16$(not in LMFDB)
6.2.e_m_y_bu_cu_ei$16$(not in LMFDB)
6.2.ae_e_e_ag_aq_bs$24$(not in LMFDB)
6.2.ae_i_am_s_abg_ca$24$(not in LMFDB)
6.2.a_a_ae_c_a_m$24$(not in LMFDB)
6.2.a_a_e_c_a_m$24$(not in LMFDB)
6.2.a_e_ae_k_aq_u$24$(not in LMFDB)
6.2.a_e_e_k_q_u$24$(not in LMFDB)
6.2.e_e_ae_ag_q_bs$24$(not in LMFDB)
6.2.e_i_m_s_bg_ca$24$(not in LMFDB)
6.2.ac_ac_e_g_ae_am$48$(not in LMFDB)
6.2.ac_g_am_w_abk_ca$48$(not in LMFDB)
6.2.a_ag_a_w_a_aca$48$(not in LMFDB)
6.2.a_ac_a_g_a_am$48$(not in LMFDB)
6.2.a_c_a_g_a_m$48$(not in LMFDB)
6.2.a_g_a_w_a_ca$48$(not in LMFDB)
6.2.c_ac_ae_g_e_am$48$(not in LMFDB)
6.2.c_g_m_w_bk_ca$48$(not in LMFDB)