Properties

Label 4.3.ak_bx_afu_ma
Base Field $\F_{3}$
Dimension $4$
Ordinary No
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian No

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Invariants

Base field:  $\F_{3}$
Dimension:  $4$
L-polynomial:  $( 1 - 3 x + 3 x^{2} )^{2}( 1 - 2 x + 3 x^{2} )^{2}$
Frobenius angles:  $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.304086723985$, $\pm0.304086723985$
Angle rank:  $1$ (numerical)

This isogeny class is not simple.

Newton polygon

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

Point counts

This isogeny class is principally polarizable, but does not contain a Jacobian.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 4 7056 1132096 76317696 4300998724 287570497536 22747538835844 1880566937174016 153037296038433856 12256289074986008976

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -6 8 48 128 294 746 2178 6656 20064 59528

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ad 2 $\times$ 1.3.ac 2 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 $\overline{\F}_{3}$
The base change of $A$ to $\F_{3^{6}}$ is 1.729.abu 2 $\times$ 1.729.cc 2 . The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{3^{6}}$.
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
4.3.ag_r_abe_bw$2$(not in LMFDB)
4.3.ae_h_a_am$2$(not in LMFDB)
4.3.ac_b_ag_y$2$(not in LMFDB)
4.3.a_ab_a_m$2$(not in LMFDB)
4.3.c_b_g_y$2$(not in LMFDB)
4.3.e_h_a_am$2$(not in LMFDB)
4.3.g_r_be_bw$2$(not in LMFDB)
4.3.k_bx_fu_ma$2$(not in LMFDB)
4.3.ah_bc_acx_fu$3$(not in LMFDB)
4.3.ae_e_m_abn$3$(not in LMFDB)
4.3.ae_q_abk_da$3$(not in LMFDB)
4.3.ab_b_g_am$3$(not in LMFDB)
4.3.ab_e_d_g$3$(not in LMFDB)
4.3.c_ac_a_p$3$(not in LMFDB)
4.3.c_h_s_y$3$(not in LMFDB)
4.3.f_n_be_ci$3$(not in LMFDB)
4.3.i_bc_ci_eb$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
4.3.ag_r_abe_bw$2$(not in LMFDB)
4.3.ae_h_a_am$2$(not in LMFDB)
4.3.ac_b_ag_y$2$(not in LMFDB)
4.3.a_ab_a_m$2$(not in LMFDB)
4.3.c_b_g_y$2$(not in LMFDB)
4.3.e_h_a_am$2$(not in LMFDB)
4.3.g_r_be_bw$2$(not in LMFDB)
4.3.k_bx_fu_ma$2$(not in LMFDB)
4.3.ah_bc_acx_fu$3$(not in LMFDB)
4.3.ae_e_m_abn$3$(not in LMFDB)
4.3.ae_q_abk_da$3$(not in LMFDB)
4.3.ab_b_g_am$3$(not in LMFDB)
4.3.ab_e_d_g$3$(not in LMFDB)
4.3.c_ac_a_p$3$(not in LMFDB)
4.3.c_h_s_y$3$(not in LMFDB)
4.3.f_n_be_ci$3$(not in LMFDB)
4.3.i_bc_ci_eb$3$(not in LMFDB)
4.3.ag_n_ag_am$4$(not in LMFDB)
4.3.ae_n_ay_bw$4$(not in LMFDB)
4.3.a_af_a_y$4$(not in LMFDB)
4.3.a_b_a_m$4$(not in LMFDB)
4.3.a_f_a_y$4$(not in LMFDB)
4.3.e_n_y_bw$4$(not in LMFDB)
4.3.g_n_g_am$4$(not in LMFDB)
4.3.ai_bc_aci_eb$6$(not in LMFDB)
4.3.af_n_abe_ci$6$(not in LMFDB)
4.3.ad_i_ap_be$6$(not in LMFDB)
4.3.ac_ac_a_p$6$(not in LMFDB)
4.3.ac_h_as_y$6$(not in LMFDB)
4.3.a_i_a_be$6$(not in LMFDB)
4.3.b_b_ag_am$6$(not in LMFDB)
4.3.b_e_ad_g$6$(not in LMFDB)
4.3.d_i_p_be$6$(not in LMFDB)
4.3.e_e_am_abn$6$(not in LMFDB)
4.3.e_q_bk_da$6$(not in LMFDB)
4.3.h_bc_cx_fu$6$(not in LMFDB)
4.3.ak_bv_afi_kw$8$(not in LMFDB)
4.3.ae_f_a_ag$8$(not in LMFDB)
4.3.ae_l_ay_bq$8$(not in LMFDB)
4.3.ac_ab_g_ag$8$(not in LMFDB)
4.3.c_ab_ag_ag$8$(not in LMFDB)
4.3.e_f_a_ag$8$(not in LMFDB)
4.3.e_l_y_bq$8$(not in LMFDB)
4.3.k_bv_fi_kw$8$(not in LMFDB)
4.3.ae_e_m_abq$12$(not in LMFDB)
4.3.ad_e_ad_g$12$(not in LMFDB)
4.3.ac_af_g_m$12$(not in LMFDB)
4.3.ac_e_am_v$12$(not in LMFDB)
4.3.a_ai_a_be$12$(not in LMFDB)
4.3.a_ae_a_g$12$(not in LMFDB)
4.3.a_e_a_g$12$(not in LMFDB)
4.3.c_af_ag_m$12$(not in LMFDB)
4.3.c_e_m_v$12$(not in LMFDB)
4.3.d_e_d_g$12$(not in LMFDB)
4.3.e_e_am_abq$12$(not in LMFDB)
4.3.ah_ba_acr_fi$24$(not in LMFDB)
4.3.ae_c_m_abe$24$(not in LMFDB)
4.3.ae_i_am_s$24$(not in LMFDB)
4.3.ae_k_am_s$24$(not in LMFDB)
4.3.ae_o_abk_co$24$(not in LMFDB)
4.3.ac_b_ag_s$24$(not in LMFDB)
4.3.ab_c_ad_ag$24$(not in LMFDB)
4.3.a_ac_a_s$24$(not in LMFDB)
4.3.a_c_a_s$24$(not in LMFDB)
4.3.b_c_d_ag$24$(not in LMFDB)
4.3.c_b_g_s$24$(not in LMFDB)
4.3.e_c_am_abe$24$(not in LMFDB)
4.3.e_i_m_s$24$(not in LMFDB)
4.3.e_k_m_s$24$(not in LMFDB)
4.3.e_o_bk_co$24$(not in LMFDB)
4.3.h_ba_cr_fi$24$(not in LMFDB)