Properties

Label 4.4.al_ch_aht_si
Base field $\F_{2^{2}}$
Dimension $4$
$p$-rank $3$
Ordinary no
Supersingular no
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:  $( 1 - 2 x )^{2}( 1 - x + 4 x^{2} )( 1 - 3 x + 4 x^{2} )^{2}$
  $1 - 11 x + 59 x^{2} - 201 x^{3} + 476 x^{4} - 804 x^{5} + 944 x^{6} - 704 x^{7} + 256 x^{8}$
Frobenius angles:  $0$, $0$, $\pm0.230053456163$, $\pm0.230053456163$, $\pm0.419569376745$
Angle rank:  $2$ (numerical)

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $16$ $55296$ $20392624$ $4478976000$ $1084563396496$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-6$ $14$ $78$ $270$ $1014$ $4070$ $16206$ $64350$ $258342$ $1042454$

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

Endomorphism algebra over $\F_{2^{2}}$
The isogeny class factors as 1.4.ae $\times$ 1.4.ad 2 $\times$ 1.4.ab and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

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.aj_bn_aeh_jk$2$(not in LMFDB)
4.4.af_l_ap_u$2$(not in LMFDB)
4.4.ad_d_aj_bc$2$(not in LMFDB)
4.4.ad_d_p_abs$2$(not in LMFDB)
4.4.ab_ab_j_ae$2$(not in LMFDB)
4.4.b_ab_aj_ae$2$(not in LMFDB)
4.4.d_d_ap_abs$2$(not in LMFDB)
4.4.d_d_j_bc$2$(not in LMFDB)
4.4.f_l_p_u$2$(not in LMFDB)
4.4.j_bn_eh_jk$2$(not in LMFDB)
4.4.l_ch_ht_si$2$(not in LMFDB)
4.4.af_r_abn_di$3$(not in LMFDB)
4.4.ac_c_d_abc$3$(not in LMFDB)
4.4.e_o_bn_di$3$(not in LMFDB)

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

TwistExtension degreeCommon base change
4.4.aj_bn_aeh_jk$2$(not in LMFDB)
4.4.af_l_ap_u$2$(not in LMFDB)
4.4.ad_d_aj_bc$2$(not in LMFDB)
4.4.ad_d_p_abs$2$(not in LMFDB)
4.4.ab_ab_j_ae$2$(not in LMFDB)
4.4.b_ab_aj_ae$2$(not in LMFDB)
4.4.d_d_ap_abs$2$(not in LMFDB)
4.4.d_d_j_bc$2$(not in LMFDB)
4.4.f_l_p_u$2$(not in LMFDB)
4.4.j_bn_eh_jk$2$(not in LMFDB)
4.4.l_ch_ht_si$2$(not in LMFDB)
4.4.af_r_abn_di$3$(not in LMFDB)
4.4.ac_c_d_abc$3$(not in LMFDB)
4.4.e_o_bn_di$3$(not in LMFDB)
4.4.ah_bf_adp_ii$4$(not in LMFDB)
4.4.af_n_az_bs$4$(not in LMFDB)
4.4.af_t_abz_eq$4$(not in LMFDB)
4.4.ad_f_ap_bk$4$(not in LMFDB)
4.4.ab_h_ad_y$4$(not in LMFDB)
4.4.ab_j_af_bo$4$(not in LMFDB)
4.4.b_h_d_y$4$(not in LMFDB)
4.4.b_j_f_bo$4$(not in LMFDB)
4.4.d_f_p_bk$4$(not in LMFDB)
4.4.f_n_z_bs$4$(not in LMFDB)
4.4.f_t_bz_eq$4$(not in LMFDB)
4.4.h_bf_dp_ii$4$(not in LMFDB)
4.4.aj_bt_afr_ni$6$(not in LMFDB)
4.4.ai_bg_adp_ie$6$(not in LMFDB)
4.4.ah_bd_add_ha$6$(not in LMFDB)
4.4.ag_s_abz_eu$6$(not in LMFDB)
4.4.ag_y_acr_fy$6$(not in LMFDB)
4.4.ae_o_abn_di$6$(not in LMFDB)
4.4.ad_j_av_cg$6$(not in LMFDB)
4.4.ad_j_aj_w$6$(not in LMFDB)
4.4.ac_i_av_bm$6$(not in LMFDB)
4.4.ab_f_ad_ba$6$(not in LMFDB)
4.4.a_a_ad_au$6$(not in LMFDB)
4.4.a_a_d_au$6$(not in LMFDB)
4.4.a_g_ap_k$6$(not in LMFDB)
4.4.a_g_p_k$6$(not in LMFDB)
4.4.b_f_d_ba$6$(not in LMFDB)
4.4.c_c_ad_abc$6$(not in LMFDB)
4.4.c_i_v_bm$6$(not in LMFDB)
4.4.d_j_j_w$6$(not in LMFDB)
4.4.d_j_v_cg$6$(not in LMFDB)
4.4.f_r_bn_di$6$(not in LMFDB)
4.4.g_s_bz_eu$6$(not in LMFDB)
4.4.g_y_cr_fy$6$(not in LMFDB)
4.4.h_bd_dd_ha$6$(not in LMFDB)
4.4.i_bg_dp_ie$6$(not in LMFDB)
4.4.j_bt_fr_ni$6$(not in LMFDB)
4.4.ae_q_abt_ds$12$(not in LMFDB)
4.4.ad_l_ap_bq$12$(not in LMFDB)
4.4.ac_k_abb_bw$12$(not in LMFDB)
4.4.ab_h_af_bm$12$(not in LMFDB)
4.4.b_h_f_bm$12$(not in LMFDB)
4.4.c_k_bb_bw$12$(not in LMFDB)
4.4.d_l_p_bq$12$(not in LMFDB)
4.4.e_q_bt_ds$12$(not in LMFDB)