Properties

Label 4.4.ak_bs_aeq_jw
Base field $\F_{2^{2}}$
Dimension $4$
$p$-rank $0$
Ordinary no
Supersingular yes
Simple no
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian no

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Invariants

Base field:  $\F_{2^{2}}$
Dimension:  $4$
L-polynomial:  $( 1 - 2 x )^{4}( 1 - 2 x + 4 x^{2} - 8 x^{3} + 16 x^{4} )$
  $1 - 10 x + 44 x^{2} - 120 x^{3} + 256 x^{4} - 480 x^{5} + 704 x^{6} - 640 x^{7} + 256 x^{8}$
Frobenius angles:  $0$, $0$, $0$, $0$, $\pm0.200000000000$, $\pm0.600000000000$
Angle rank:  $0$ (numerical)

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:  $[1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $11$ $27621$ $8742041$ $3538940625$ $1095222947841$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-5$ $5$ $25$ $209$ $1025$ $3905$ $15745$ $64769$ $259585$ $1040385$

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 $\F_{2^{20}}$.

Endomorphism algebra over $\F_{2^{2}}$
The isogeny class factors as 1.4.ae 2 $\times$ 2.4.ac_e 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 $\overline{\F}_{2^{2}}$
The base change of $A$ to $\F_{2^{20}}$ is 1.1048576.adau 4 and its endomorphism algebra is $\mathrm{M}_{4}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
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
4.4.ag_m_ai_a$2$(not in LMFDB)
4.4.ac_ae_i_a$2$(not in LMFDB)
4.4.c_ae_ai_a$2$(not in LMFDB)
4.4.g_m_i_a$2$(not in LMFDB)
4.4.k_bs_eq_jw$2$(not in LMFDB)
4.4.ae_i_ay_cm$3$(not in LMFDB)
4.4.c_i_a_q$3$(not in LMFDB)

Below is a list of all twists of this isogeny class.

TwistExtension degreeCommon base change
4.4.ag_m_ai_a$2$(not in LMFDB)
4.4.ac_ae_i_a$2$(not in LMFDB)
4.4.c_ae_ai_a$2$(not in LMFDB)
4.4.g_m_i_a$2$(not in LMFDB)
4.4.k_bs_eq_jw$2$(not in LMFDB)
4.4.ae_i_ay_cm$3$(not in LMFDB)
4.4.c_i_a_q$3$(not in LMFDB)
4.4.ag_u_ace_ey$4$(not in LMFDB)
4.4.ac_e_ai_a$4$(not in LMFDB)
4.4.ac_m_ay_cm$4$(not in LMFDB)
4.4.c_e_i_a$4$(not in LMFDB)
4.4.c_m_y_cm$4$(not in LMFDB)
4.4.g_u_ce_ey$4$(not in LMFDB)
4.4.a_aq_a_ds$5$(not in LMFDB)
4.4.a_e_a_q$5$(not in LMFDB)
4.4.k_bs_eq_jw$5$(not in LMFDB)
4.4.ai_bg_adk_hk$6$(not in LMFDB)
4.4.ag_y_acm_fo$6$(not in LMFDB)
4.4.ae_i_ai_a$6$(not in LMFDB)
4.4.ac_i_aq_bw$6$(not in LMFDB)
4.4.ac_i_a_q$6$(not in LMFDB)
4.4.a_a_ai_a$6$(not in LMFDB)
4.4.a_a_i_a$6$(not in LMFDB)
4.4.c_i_q_bw$6$(not in LMFDB)
4.4.e_i_i_a$6$(not in LMFDB)
4.4.e_i_y_cm$6$(not in LMFDB)
4.4.g_y_cm_fo$6$(not in LMFDB)
4.4.i_bg_dk_hk$6$(not in LMFDB)
4.4.ac_e_ai_bg$8$(not in LMFDB)
4.4.c_e_i_bg$8$(not in LMFDB)
4.4.aq_ei_arg_brc$10$(not in LMFDB)
4.4.ai_q_bg_age$10$(not in LMFDB)
4.4.ae_m_abg_dc$10$(not in LMFDB)
4.4.e_m_bg_dc$10$(not in LMFDB)
4.4.i_q_abg_age$10$(not in LMFDB)
4.4.q_ei_rg_brc$10$(not in LMFDB)
4.4.ae_q_abo_ds$12$(not in LMFDB)
4.4.ac_a_a_q$12$(not in LMFDB)
4.4.a_i_ai_bg$12$(not in LMFDB)
4.4.a_i_i_bg$12$(not in LMFDB)
4.4.c_a_a_q$12$(not in LMFDB)
4.4.e_q_bo_ds$12$(not in LMFDB)
4.4.am_cq_ajg_we$15$(not in LMFDB)
4.4.ag_i_y_ads$15$(not in LMFDB)
4.4.ag_u_abw_ds$15$(not in LMFDB)
4.4.ac_i_a_q$15$(not in LMFDB)
4.4.a_ae_a_a$15$(not in LMFDB)
4.4.a_i_a_bw$15$(not in LMFDB)
4.4.e_i_y_cm$15$(not in LMFDB)
4.4.g_i_ay_ads$15$(not in LMFDB)
4.4.g_u_bw_ds$15$(not in LMFDB)
4.4.m_cq_jg_we$15$(not in LMFDB)
4.4.am_cm_aia_sm$20$(not in LMFDB)
4.4.ai_bg_ads_iq$20$(not in LMFDB)
4.4.ae_a_q_abg$20$(not in LMFDB)
4.4.ae_q_abw_ds$20$(not in LMFDB)
4.4.a_ae_a_q$20$(not in LMFDB)
4.4.a_a_a_abg$20$(not in LMFDB)
4.4.a_q_a_ds$20$(not in LMFDB)
4.4.e_a_aq_abg$20$(not in LMFDB)
4.4.e_q_bw_ds$20$(not in LMFDB)
4.4.i_bg_ds_iq$20$(not in LMFDB)
4.4.m_cm_ia_sm$20$(not in LMFDB)
4.4.ao_dk_amq_beu$30$(not in LMFDB)
4.4.ak_bo_adk_ge$30$(not in LMFDB)
4.4.ak_ca_agu_qa$30$(not in LMFDB)
4.4.ai_bc_acm_ey$30$(not in LMFDB)
4.4.ai_bo_aey_ls$30$(not in LMFDB)
4.4.ae_e_aq_cm$30$(not in LMFDB)
4.4.ae_e_q_acm$30$(not in LMFDB)
4.4.ae_q_abg_dc$30$(not in LMFDB)
4.4.ac_ai_i_bg$30$(not in LMFDB)
4.4.ac_a_i_aq$30$(not in LMFDB)
4.4.ac_e_aq_bg$30$(not in LMFDB)
4.4.ac_e_q_abg$30$(not in LMFDB)
4.4.c_ai_ai_bg$30$(not in LMFDB)
4.4.c_a_ai_aq$30$(not in LMFDB)
4.4.c_e_aq_abg$30$(not in LMFDB)
4.4.c_e_q_bg$30$(not in LMFDB)
4.4.e_e_aq_acm$30$(not in LMFDB)
4.4.e_e_q_cm$30$(not in LMFDB)
4.4.e_q_bg_dc$30$(not in LMFDB)
4.4.i_bc_cm_ey$30$(not in LMFDB)
4.4.i_bo_ey_ls$30$(not in LMFDB)
4.4.k_bo_dk_ge$30$(not in LMFDB)
4.4.k_ca_gu_qa$30$(not in LMFDB)
4.4.o_dk_mq_beu$30$(not in LMFDB)
4.4.ai_y_abg_bg$40$(not in LMFDB)
4.4.ae_i_aq_bg$40$(not in LMFDB)
4.4.a_ai_a_bg$40$(not in LMFDB)
4.4.a_a_a_bg$40$(not in LMFDB)
4.4.a_i_a_bg$40$(not in LMFDB)
4.4.e_i_q_bg$40$(not in LMFDB)
4.4.i_y_bg_bg$40$(not in LMFDB)
4.4.ak_bw_afw_no$60$(not in LMFDB)
4.4.ai_u_a_acm$60$(not in LMFDB)
4.4.ai_bk_aei_jw$60$(not in LMFDB)
4.4.ag_m_a_abg$60$(not in LMFDB)
4.4.ag_q_abo_ds$60$(not in LMFDB)
4.4.ag_y_acu_ge$60$(not in LMFDB)
4.4.ag_bc_adc_hk$60$(not in LMFDB)
4.4.ae_e_a_a$60$(not in LMFDB)
4.4.ae_i_a_aq$60$(not in LMFDB)
4.4.ae_m_abg_cm$60$(not in LMFDB)
4.4.ae_u_abw_ey$60$(not in LMFDB)
4.4.ac_ae_a_bg$60$(not in LMFDB)
4.4.ac_a_i_abg$60$(not in LMFDB)
4.4.ac_e_a_a$60$(not in LMFDB)
4.4.ac_i_ay_bg$60$(not in LMFDB)
4.4.ac_m_aq_cm$60$(not in LMFDB)
4.4.ac_q_ay_ds$60$(not in LMFDB)
4.4.a_am_a_cm$60$(not in LMFDB)
4.4.a_ai_a_bw$60$(not in LMFDB)
4.4.a_a_a_q$60$(not in LMFDB)
4.4.a_e_aq_a$60$(not in LMFDB)
4.4.a_e_a_a$60$(not in LMFDB)
4.4.a_e_q_a$60$(not in LMFDB)
4.4.a_m_a_cm$60$(not in LMFDB)
4.4.c_ae_a_bg$60$(not in LMFDB)
4.4.c_a_ai_abg$60$(not in LMFDB)
4.4.c_e_a_a$60$(not in LMFDB)
4.4.c_i_y_bg$60$(not in LMFDB)
4.4.c_m_q_cm$60$(not in LMFDB)
4.4.c_q_y_ds$60$(not in LMFDB)
4.4.e_e_a_a$60$(not in LMFDB)
4.4.e_i_a_aq$60$(not in LMFDB)
4.4.e_m_bg_cm$60$(not in LMFDB)
4.4.e_u_bw_ey$60$(not in LMFDB)
4.4.g_m_a_abg$60$(not in LMFDB)
4.4.g_q_bo_ds$60$(not in LMFDB)
4.4.g_y_cu_ge$60$(not in LMFDB)
4.4.g_bc_dc_hk$60$(not in LMFDB)
4.4.i_u_a_acm$60$(not in LMFDB)
4.4.i_bk_ei_jw$60$(not in LMFDB)
4.4.k_bw_fw_no$60$(not in LMFDB)
4.4.a_a_a_a$80$(not in LMFDB)
4.4.ae_e_ai_bg$90$(not in LMFDB)
4.4.ae_e_i_abg$90$(not in LMFDB)
4.4.ac_e_ai_q$90$(not in LMFDB)
4.4.ac_e_i_aq$90$(not in LMFDB)
4.4.c_e_ai_aq$90$(not in LMFDB)
4.4.c_e_i_q$90$(not in LMFDB)
4.4.e_e_ai_abg$90$(not in LMFDB)
4.4.e_e_i_bg$90$(not in LMFDB)
4.4.ag_q_ay_bg$120$(not in LMFDB)
4.4.ae_m_aq_bg$120$(not in LMFDB)
4.4.ac_a_ai_bg$120$(not in LMFDB)
4.4.ac_i_ai_bg$120$(not in LMFDB)
4.4.a_ae_a_bg$120$(not in LMFDB)
4.4.a_a_a_aq$120$(not in LMFDB)
4.4.a_e_a_bg$120$(not in LMFDB)
4.4.c_a_i_bg$120$(not in LMFDB)
4.4.c_i_i_bg$120$(not in LMFDB)
4.4.e_m_q_bg$120$(not in LMFDB)
4.4.g_q_y_bg$120$(not in LMFDB)
4.4.a_e_ai_a$180$(not in LMFDB)
4.4.a_e_i_a$180$(not in LMFDB)