Properties

Label 4.4.am_cn_aii_tk
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

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Invariants

Base field:  $\F_{2^{2}}$
Dimension:  $4$
L-polynomial:  $( 1 - 2 x )^{4}( 1 - 4 x + 9 x^{2} - 16 x^{3} + 16 x^{4} )$
  $1 - 12 x + 65 x^{2} - 216 x^{3} + 504 x^{4} - 864 x^{5} + 1040 x^{6} - 768 x^{7} + 256 x^{8}$
Frobenius angles:  $0$, $0$, $0$, $0$, $\pm0.117169895439$, $\pm0.478661301576$
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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $6$ $22356$ $9464742$ $2906280000$ $966156270486$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-7$ $3$ $29$ $159$ $893$ $4035$ $16205$ $64575$ $260525$ $1047843$

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.ae_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.ae_b_i_ai$2$(not in LMFDB)
4.4.ae_b_q_abo$2$(not in LMFDB)
4.4.e_b_aq_abo$2$(not in LMFDB)
4.4.e_b_ai_ai$2$(not in LMFDB)
4.4.m_cn_ii_tk$2$(not in LMFDB)
4.4.ag_r_abq_ds$3$(not in LMFDB)
4.4.a_f_am_m$3$(not in LMFDB)

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

TwistExtension degreeCommon base change
4.4.ae_b_i_ai$2$(not in LMFDB)
4.4.ae_b_q_abo$2$(not in LMFDB)
4.4.e_b_aq_abo$2$(not in LMFDB)
4.4.e_b_ai_ai$2$(not in LMFDB)
4.4.m_cn_ii_tk$2$(not in LMFDB)
4.4.ag_r_abq_ds$3$(not in LMFDB)
4.4.a_f_am_m$3$(not in LMFDB)
4.4.ai_bh_adw_iy$4$(not in LMFDB)
4.4.ae_r_abw_ea$4$(not in LMFDB)
4.4.a_b_ae_ay$4$(not in LMFDB)
4.4.a_b_e_ay$4$(not in LMFDB)
4.4.e_r_bw_ea$4$(not in LMFDB)
4.4.i_bh_dw_iy$4$(not in LMFDB)
4.4.ac_f_ag_e$5$(not in LMFDB)
4.4.ak_bx_agc_oe$6$(not in LMFDB)
4.4.ai_bl_aem_ki$6$(not in LMFDB)
4.4.ae_n_abg_cq$6$(not in LMFDB)
4.4.ac_b_c_aq$6$(not in LMFDB)
4.4.ac_b_k_abg$6$(not in LMFDB)
4.4.a_f_m_m$6$(not in LMFDB)
4.4.c_b_ak_abg$6$(not in LMFDB)
4.4.c_b_ac_aq$6$(not in LMFDB)
4.4.e_n_bg_cq$6$(not in LMFDB)
4.4.g_r_bq_ds$6$(not in LMFDB)
4.4.i_bl_em_ki$6$(not in LMFDB)
4.4.k_bx_gc_oe$6$(not in LMFDB)
4.4.ae_j_aq_bg$8$(not in LMFDB)
4.4.e_j_q_bg$8$(not in LMFDB)
4.4.ag_v_acg_fc$10$(not in LMFDB)
4.4.c_f_g_e$10$(not in LMFDB)
4.4.g_v_cg_fc$10$(not in LMFDB)
4.4.ag_z_acw_gm$12$(not in LMFDB)
4.4.ae_f_a_ae$12$(not in LMFDB)
4.4.ac_j_aw_bo$12$(not in LMFDB)
4.4.c_j_w_bo$12$(not in LMFDB)
4.4.e_f_a_ae$12$(not in LMFDB)
4.4.g_z_cw_gm$12$(not in LMFDB)