Properties

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

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Invariants

Base field:  $\F_{3}$
Dimension:  $4$
L-polynomial:  $( 1 - 3 x + 3 x^{2} )( 1 - x + 3 x^{2} )( 1 - 4 x + 8 x^{2} - 12 x^{3} + 9 x^{4} )$
Frobenius angles:  $\pm0.0540867239847$, $\pm0.166666666667$, $\pm0.406785250661$, $\pm0.445913276015$
Angle rank:  $2$ (numerical)

This isogeny class is not simple.

Newton polygon

$p$-rank:  $3$
Slopes:  $[0, 0, 0, 1/2, 1/2, 1, 1, 1]$

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
$A(\F_{q^r})$ 6 7140 631008 31558800 2844704886 300359808000 25239404286102 1889625817497600 146877530321315808 12011040839153825700

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -4 12 32 56 196 774 2404 6688 19256 58332

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ad $\times$ 1.3.ab $\times$ 2.3.ae_i 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 $\F_{3^{12}}$ is 1.531441.acec $\times$ 1.531441.zi 2 $\times$ 1.531441.cag. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{3^{12}}$.
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_t_abu_dm$2$(not in LMFDB)
4.3.ac_d_ac_ag$2$(not in LMFDB)
4.3.a_b_ae_ag$2$(not in LMFDB)
4.3.a_b_e_ag$2$(not in LMFDB)
4.3.c_d_c_ag$2$(not in LMFDB)
4.3.g_t_bu_dm$2$(not in LMFDB)
4.3.i_bh_do_he$2$(not in LMFDB)
4.3.af_s_abv_dm$3$(not in LMFDB)
4.3.ac_d_ac_ag$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
4.3.ag_t_abu_dm$2$(not in LMFDB)
4.3.ac_d_ac_ag$2$(not in LMFDB)
4.3.a_b_ae_ag$2$(not in LMFDB)
4.3.a_b_e_ag$2$(not in LMFDB)
4.3.c_d_c_ag$2$(not in LMFDB)
4.3.g_t_bu_dm$2$(not in LMFDB)
4.3.i_bh_do_he$2$(not in LMFDB)
4.3.af_s_abv_dm$3$(not in LMFDB)
4.3.ac_d_ac_ag$3$(not in LMFDB)
4.3.ad_k_az_bq$6$(not in LMFDB)
4.3.d_k_z_bq$6$(not in LMFDB)
4.3.f_s_bv_dm$6$(not in LMFDB)
4.3.ai_bj_adw_hw$8$(not in LMFDB)
4.3.ag_v_aby_ds$8$(not in LMFDB)
4.3.ae_h_ae_a$8$(not in LMFDB)
4.3.ae_l_au_bk$8$(not in LMFDB)
4.3.ac_b_ac_m$8$(not in LMFDB)
4.3.ac_f_ak_y$8$(not in LMFDB)
4.3.ac_f_c_a$8$(not in LMFDB)
4.3.a_d_ae_m$8$(not in LMFDB)
4.3.a_d_e_m$8$(not in LMFDB)
4.3.c_b_c_m$8$(not in LMFDB)
4.3.c_f_ac_a$8$(not in LMFDB)
4.3.c_f_k_y$8$(not in LMFDB)
4.3.e_h_e_a$8$(not in LMFDB)
4.3.e_l_u_bk$8$(not in LMFDB)
4.3.g_v_by_ds$8$(not in LMFDB)
4.3.i_bj_dw_hw$8$(not in LMFDB)
4.3.ag_s_abo_cx$24$(not in LMFDB)
4.3.af_u_abx_dy$24$(not in LMFDB)
4.3.ae_i_au_bt$24$(not in LMFDB)
4.3.ad_j_aw_bk$24$(not in LMFDB)
4.3.ad_m_ax_cc$24$(not in LMFDB)
4.3.ac_c_i_av$24$(not in LMFDB)
4.3.ab_e_ab_g$24$(not in LMFDB)
4.3.ab_f_ao_m$24$(not in LMFDB)
4.3.ab_i_af_be$24$(not in LMFDB)
4.3.a_a_ae_ad$24$(not in LMFDB)
4.3.a_a_e_ad$24$(not in LMFDB)
4.3.b_e_b_g$24$(not in LMFDB)
4.3.b_f_o_m$24$(not in LMFDB)
4.3.b_i_f_be$24$(not in LMFDB)
4.3.c_c_ai_av$24$(not in LMFDB)
4.3.d_j_w_bk$24$(not in LMFDB)
4.3.d_m_x_cc$24$(not in LMFDB)
4.3.e_i_u_bt$24$(not in LMFDB)
4.3.f_u_bx_dy$24$(not in LMFDB)
4.3.g_s_bo_cx$24$(not in LMFDB)