Properties

Label 4.4.ak_bv_afo_mq
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:  $( 1 - 2 x )^{4}( 1 - 2 x + 7 x^{2} - 8 x^{3} + 16 x^{4} )$
  $1 - 10 x + 47 x^{2} - 144 x^{3} + 328 x^{4} - 576 x^{5} + 752 x^{6} - 640 x^{7} + 256 x^{8}$
Frobenius angles:  $0$, $0$, $0$, $0$, $\pm0.293751018564$, $\pm0.533021264456$
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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $14$ $38556$ $11529602$ $3277260000$ $1006429174254$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-5$ $11$ $43$ $191$ $935$ $3851$ $15451$ $63935$ $261367$ $1047531$

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.ac_h 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.ag_p_abg_cu$2$(not in LMFDB)
4.4.ac_ab_i_ay$2$(not in LMFDB)
4.4.c_ab_ai_ay$2$(not in LMFDB)
4.4.g_p_bg_cu$2$(not in LMFDB)
4.4.k_bv_fo_mq$2$(not in LMFDB)
4.4.ae_l_abe_cm$3$(not in LMFDB)
4.4.c_l_m_ca$3$(not in LMFDB)

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

TwistExtension degreeCommon base change
4.4.ag_p_abg_cu$2$(not in LMFDB)
4.4.ac_ab_i_ay$2$(not in LMFDB)
4.4.c_ab_ai_ay$2$(not in LMFDB)
4.4.g_p_bg_cu$2$(not in LMFDB)
4.4.k_bv_fo_mq$2$(not in LMFDB)
4.4.ae_l_abe_cm$3$(not in LMFDB)
4.4.c_l_m_ca$3$(not in LMFDB)
4.4.ag_x_acq_fw$4$(not in LMFDB)
4.4.ac_h_au_y$4$(not in LMFDB)
4.4.ac_p_ay_dk$4$(not in LMFDB)
4.4.c_h_u_y$4$(not in LMFDB)
4.4.c_p_y_dk$4$(not in LMFDB)
4.4.g_x_cq_fw$4$(not in LMFDB)
4.4.a_h_g_bc$5$(not in LMFDB)
4.4.ai_bj_aec_jg$6$(not in LMFDB)
4.4.ag_bb_acy_gy$6$(not in LMFDB)
4.4.ae_l_aba_bw$6$(not in LMFDB)
4.4.ac_l_aq_ci$6$(not in LMFDB)
4.4.ac_l_am_ca$6$(not in LMFDB)
4.4.a_d_ao_a$6$(not in LMFDB)
4.4.a_d_o_a$6$(not in LMFDB)
4.4.c_l_q_ci$6$(not in LMFDB)
4.4.e_l_ba_bw$6$(not in LMFDB)
4.4.e_l_be_cm$6$(not in LMFDB)
4.4.g_bb_cy_gy$6$(not in LMFDB)
4.4.i_bj_ec_jg$6$(not in LMFDB)
4.4.ac_h_ai_bg$8$(not in LMFDB)
4.4.c_h_i_bg$8$(not in LMFDB)
4.4.ae_p_abm_do$10$(not in LMFDB)
4.4.a_h_ag_bc$10$(not in LMFDB)
4.4.e_p_bm_do$10$(not in LMFDB)
4.4.ae_t_abu_eq$12$(not in LMFDB)
4.4.ac_d_a_e$12$(not in LMFDB)
4.4.a_l_ac_ce$12$(not in LMFDB)
4.4.a_l_c_ce$12$(not in LMFDB)
4.4.c_d_a_e$12$(not in LMFDB)
4.4.e_t_bu_eq$12$(not in LMFDB)