Properties

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

Downloads

Learn more

Invariants

Base field:  $\F_{2^{2}}$
Dimension:  $4$
L-polynomial:  $( 1 - 2 x )^{4}( 1 - 3 x + 9 x^{2} - 12 x^{3} + 16 x^{4} )$
  $1 - 11 x + 57 x^{2} - 188 x^{3} + 440 x^{4} - 752 x^{5} + 912 x^{6} - 704 x^{7} + 256 x^{8}$
Frobenius angles:  $0$, $0$, $0$, $0$, $\pm0.272875599394$, $\pm0.469557725221$
Angle rank:  $2$ (numerical)

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $11$ $36531$ $13099856$ $3310621875$ $966715000691$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-6$ $10$ $51$ $194$ $894$ $3895$ $15786$ $63714$ $259179$ $1047010$

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 2 $\times$ 2.4.ad_j 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.af_j_au_ce$2$(not in LMFDB)
4.4.ad_b_m_abo$2$(not in LMFDB)
4.4.d_b_am_abo$2$(not in LMFDB)
4.4.f_j_u_ce$2$(not in LMFDB)
4.4.l_cf_hg_qy$2$(not in LMFDB)
4.4.af_p_abm_dc$3$(not in LMFDB)
4.4.b_j_e_bs$3$(not in LMFDB)

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

TwistExtension degreeCommon base change
4.4.af_j_au_ce$2$(not in LMFDB)
4.4.ad_b_m_abo$2$(not in LMFDB)
4.4.d_b_am_abo$2$(not in LMFDB)
4.4.f_j_u_ce$2$(not in LMFDB)
4.4.l_cf_hg_qy$2$(not in LMFDB)
4.4.af_p_abm_dc$3$(not in LMFDB)
4.4.b_j_e_bs$3$(not in LMFDB)
4.4.ah_bd_adk_hs$4$(not in LMFDB)
4.4.ad_r_abk_ea$4$(not in LMFDB)
4.4.ab_f_aq_i$4$(not in LMFDB)
4.4.b_f_q_i$4$(not in LMFDB)
4.4.d_r_bk_ea$4$(not in LMFDB)
4.4.h_bd_dk_hs$4$(not in LMFDB)
4.4.ab_h_c_u$5$(not in LMFDB)
4.4.aj_br_afi_mi$6$(not in LMFDB)
4.4.ah_bh_adw_jc$6$(not in LMFDB)
4.4.ad_h_as_bg$6$(not in LMFDB)
4.4.ad_n_ay_cq$6$(not in LMFDB)
4.4.ab_d_o_aq$6$(not in LMFDB)
4.4.ab_j_ae_bs$6$(not in LMFDB)
4.4.b_d_ao_aq$6$(not in LMFDB)
4.4.d_h_s_bg$6$(not in LMFDB)
4.4.d_n_y_cq$6$(not in LMFDB)
4.4.f_p_bm_dc$6$(not in LMFDB)
4.4.h_bh_dw_jc$6$(not in LMFDB)
4.4.j_br_fi_mi$6$(not in LMFDB)
4.4.ad_j_am_bg$8$(not in LMFDB)
4.4.d_j_m_bg$8$(not in LMFDB)
4.4.af_t_aby_em$10$(not in LMFDB)
4.4.b_h_ac_u$10$(not in LMFDB)
4.4.f_t_by_em$10$(not in LMFDB)
4.4.af_x_ack_fw$12$(not in LMFDB)
4.4.ad_f_a_ae$12$(not in LMFDB)
4.4.ab_l_ak_ce$12$(not in LMFDB)
4.4.b_l_k_ce$12$(not in LMFDB)
4.4.d_f_a_ae$12$(not in LMFDB)
4.4.f_x_ck_fw$12$(not in LMFDB)