Properties

Label 4.3.ah_ba_acr_fi
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:  $( 1 - 3 x + 3 x^{2} )( 1 + 3 x^{2} )( 1 - 4 x + 8 x^{2} - 12 x^{3} + 9 x^{4} )$
Frobenius angles:  $\pm0.0540867239847$, $\pm0.166666666667$, $\pm0.445913276015$, $\pm0.5$
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
$A(\F_{q^r})$ 8 7616 490784 26930176 3258722968 327058457600 24316960184056 1811776064716800 147899391592223264 12208097778913566656

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -3 13 24 45 227 838 2321 6413 19392 59293

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ad $\times$ 1.3.a $\times$ 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 2 $\times$ 1.531441.zi 2 . 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.ab_c_ad_ag$2$(not in LMFDB)
4.3.b_c_d_ag$2$(not in LMFDB)
4.3.h_ba_cr_fi$2$(not in LMFDB)
4.3.ak_bv_afi_kw$3$(not in LMFDB)
4.3.ae_f_a_ag$3$(not in LMFDB)
4.3.ae_o_abk_co$3$(not in LMFDB)
4.3.ab_c_ad_ag$3$(not in LMFDB)
4.3.c_ab_ag_ag$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
4.3.ab_c_ad_ag$2$(not in LMFDB)
4.3.b_c_d_ag$2$(not in LMFDB)
4.3.h_ba_cr_fi$2$(not in LMFDB)
4.3.ak_bv_afi_kw$3$(not in LMFDB)
4.3.ae_f_a_ag$3$(not in LMFDB)
4.3.ae_o_abk_co$3$(not in LMFDB)
4.3.ab_c_ad_ag$3$(not in LMFDB)
4.3.c_ab_ag_ag$3$(not in LMFDB)
4.3.ac_ab_g_ag$6$(not in LMFDB)
4.3.e_f_a_ag$6$(not in LMFDB)
4.3.e_o_bk_co$6$(not in LMFDB)
4.3.k_bv_fi_kw$6$(not in LMFDB)
4.3.ah_bc_acx_fu$8$(not in LMFDB)
4.3.ad_e_ad_g$8$(not in LMFDB)
4.3.ad_i_ap_be$8$(not in LMFDB)
4.3.ab_e_d_g$8$(not in LMFDB)
4.3.b_e_ad_g$8$(not in LMFDB)
4.3.d_e_d_g$8$(not in LMFDB)
4.3.d_i_p_be$8$(not in LMFDB)
4.3.h_bc_cx_fu$8$(not in LMFDB)
4.3.ae_c_m_abe$12$(not in LMFDB)
4.3.ae_l_ay_bq$12$(not in LMFDB)
4.3.e_c_am_abe$12$(not in LMFDB)
4.3.e_l_y_bq$12$(not in LMFDB)
4.3.ak_bx_afu_ma$24$(not in LMFDB)
4.3.ai_bc_aci_eb$24$(not in LMFDB)
4.3.ag_n_ag_am$24$(not in LMFDB)
4.3.ag_r_abe_bw$24$(not in LMFDB)
4.3.af_n_abe_ci$24$(not in LMFDB)
4.3.ae_e_m_abq$24$(not in LMFDB)
4.3.ae_e_m_abn$24$(not in LMFDB)
4.3.ae_h_a_am$24$(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_n_ay_bw$24$(not in LMFDB)
4.3.ae_q_abk_da$24$(not in LMFDB)
4.3.ac_af_g_m$24$(not in LMFDB)
4.3.ac_ac_a_p$24$(not in LMFDB)
4.3.ac_b_ag_s$24$(not in LMFDB)
4.3.ac_b_ag_y$24$(not in LMFDB)
4.3.ac_e_am_v$24$(not in LMFDB)
4.3.ac_h_as_y$24$(not in LMFDB)
4.3.ab_b_g_am$24$(not in LMFDB)
4.3.a_ai_a_be$24$(not in LMFDB)
4.3.a_af_a_y$24$(not in LMFDB)
4.3.a_ae_a_g$24$(not in LMFDB)
4.3.a_ac_a_s$24$(not in LMFDB)
4.3.a_ab_a_m$24$(not in LMFDB)
4.3.a_b_a_m$24$(not in LMFDB)
4.3.a_c_a_s$24$(not in LMFDB)
4.3.a_e_a_g$24$(not in LMFDB)
4.3.a_f_a_y$24$(not in LMFDB)
4.3.a_i_a_be$24$(not in LMFDB)
4.3.b_b_ag_am$24$(not in LMFDB)
4.3.c_af_ag_m$24$(not in LMFDB)
4.3.c_ac_a_p$24$(not in LMFDB)
4.3.c_b_g_s$24$(not in LMFDB)
4.3.c_b_g_y$24$(not in LMFDB)
4.3.c_e_m_v$24$(not in LMFDB)
4.3.c_h_s_y$24$(not in LMFDB)
4.3.e_e_am_abq$24$(not in LMFDB)
4.3.e_e_am_abn$24$(not in LMFDB)
4.3.e_h_a_am$24$(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_n_y_bw$24$(not in LMFDB)
4.3.e_q_bk_da$24$(not in LMFDB)
4.3.f_n_be_ci$24$(not in LMFDB)
4.3.g_n_g_am$24$(not in LMFDB)
4.3.g_r_be_bw$24$(not in LMFDB)
4.3.i_bc_ci_eb$24$(not in LMFDB)
4.3.k_bx_fu_ma$24$(not in LMFDB)