Properties

Label 4.2.af_k_aj_g
Base field $\F_{2}$
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:  $\F_{2}$
Dimension:  $4$
L-polynomial:  $( 1 - 2 x + 2 x^{2} )( 1 - 3 x + 2 x^{2} + x^{3} + 4 x^{4} - 12 x^{5} + 8 x^{6} )$
  $1 - 5 x + 10 x^{2} - 9 x^{3} + 6 x^{4} - 18 x^{5} + 40 x^{6} - 40 x^{7} + 16 x^{8}$
Frobenius angles:  $\pm0.0992589862044$, $\pm0.186455299510$, $\pm0.250000000000$, $\pm0.757883870938$
Angle rank:  $1$ (numerical)
Jacobians:  $0$
Isomorphism classes:  1

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $1$ $145$ $3913$ $203725$ $1429301$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-2$ $0$ $7$ $36$ $43$ $87$ $152$ $220$ $538$ $1015$

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 $\F_{2^{28}}$.

Endomorphism algebra over $\F_{2}$
The isogeny class factors as 1.2.ac $\times$ 3.2.ad_c_b 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 $\overline{\F}_{2}$
The base change of $A$ to $\F_{2^{28}}$ is 1.268435456.blhf 3 $\times$ 1.268435456.bwmi. The endomorphism algebra for each factor is:
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
4.2.ab_ac_ab_k$2$4.4.af_w_acj_fo
4.2.b_ac_b_k$2$4.4.af_w_acj_fo
4.2.f_k_j_g$2$4.4.af_w_acj_fo

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

TwistExtension degreeCommon base change
4.2.ab_ac_ab_k$2$4.4.af_w_acj_fo
4.2.b_ac_b_k$2$4.4.af_w_acj_fo
4.2.f_k_j_g$2$4.4.af_w_acj_fo
4.2.af_r_abl_ck$7$(not in LMFDB)
4.2.c_d_f_g$7$(not in LMFDB)
4.2.ad_e_af_i$8$(not in LMFDB)
4.2.d_e_f_i$8$(not in LMFDB)
4.2.ag_t_abp_co$14$(not in LMFDB)
4.2.ad_j_ap_ba$14$(not in LMFDB)
4.2.ac_d_af_g$14$(not in LMFDB)
4.2.ab_f_af_o$14$(not in LMFDB)
4.2.ab_f_ab_k$14$(not in LMFDB)
4.2.b_f_b_k$14$(not in LMFDB)
4.2.b_f_f_o$14$(not in LMFDB)
4.2.d_j_p_ba$14$(not in LMFDB)
4.2.f_r_bl_ck$14$(not in LMFDB)
4.2.g_t_bp_co$14$(not in LMFDB)
4.2.ac_c_f_ak$21$(not in LMFDB)
4.2.ad_d_d_ak$28$(not in LMFDB)
4.2.ab_ab_b_c$28$(not in LMFDB)
4.2.b_ab_ab_c$28$(not in LMFDB)
4.2.d_d_ad_ak$28$(not in LMFDB)
4.2.ae_i_al_o$42$(not in LMFDB)
4.2.ac_c_af_k$42$(not in LMFDB)
4.2.a_a_ad_c$42$(not in LMFDB)
4.2.a_a_d_c$42$(not in LMFDB)
4.2.c_c_af_ak$42$(not in LMFDB)
4.2.c_c_f_k$42$(not in LMFDB)
4.2.e_i_l_o$42$(not in LMFDB)
4.2.ae_l_ax_bk$56$(not in LMFDB)
4.2.ad_l_at_bk$56$(not in LMFDB)
4.2.ab_b_b_ae$56$(not in LMFDB)
4.2.ab_h_af_u$56$(not in LMFDB)
4.2.b_b_ab_ae$56$(not in LMFDB)
4.2.b_h_f_u$56$(not in LMFDB)
4.2.d_l_t_bk$56$(not in LMFDB)
4.2.e_l_x_bk$56$(not in LMFDB)
4.2.ac_e_ah_i$168$(not in LMFDB)
4.2.a_c_af_a$168$(not in LMFDB)
4.2.a_c_f_a$168$(not in LMFDB)
4.2.c_e_h_i$168$(not in LMFDB)