Properties

Label 4.3.ah_x_aby_dm
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:  $( 1 - 3 x + 3 x^{2} )( 1 - 2 x + 3 x^{2} )( 1 - 2 x + x^{2} - 6 x^{3} + 9 x^{4} )$
Frobenius angles:  $\pm0.0292466093486$, $\pm0.166666666667$, $\pm0.304086723985$, $\pm0.637420057318$
Angle rank:  $1$ (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 4788 344736 48301344 3824938986 250890587136 21939467263818 1876183595395200 148662203360249376 12107841369080712948

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -3 7 18 91 267 646 2097 6643 19494 58807

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ad $\times$ 1.3.ac $\times$ 2.3.ac_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 $\overline{\F}_{3}$
The base change of $A$ to $\F_{3^{6}}$ is 1.729.abu 3 $\times$ 1.729.cc. 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.ad_d_ak_be$2$(not in LMFDB)
4.3.ad_d_k_abe$2$(not in LMFDB)
4.3.ab_ab_ac_g$2$(not in LMFDB)
4.3.b_ab_c_g$2$(not in LMFDB)
4.3.d_d_ak_abe$2$(not in LMFDB)
4.3.d_d_k_be$2$(not in LMFDB)
4.3.h_x_by_dm$2$(not in LMFDB)
4.3.ae_l_aba_bw$3$(not in LMFDB)
4.3.ab_ab_ac_g$3$(not in LMFDB)
4.3.ab_c_af_s$3$(not in LMFDB)
4.3.c_i_k_be$3$(not in LMFDB)
4.3.f_o_z_bq$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
4.3.ad_d_ak_be$2$(not in LMFDB)
4.3.ad_d_k_abe$2$(not in LMFDB)
4.3.ab_ab_ac_g$2$(not in LMFDB)
4.3.b_ab_c_g$2$(not in LMFDB)
4.3.d_d_ak_abe$2$(not in LMFDB)
4.3.d_d_k_be$2$(not in LMFDB)
4.3.h_x_by_dm$2$(not in LMFDB)
4.3.ae_l_aba_bw$3$(not in LMFDB)
4.3.ab_ab_ac_g$3$(not in LMFDB)
4.3.ab_c_af_s$3$(not in LMFDB)
4.3.c_i_k_be$3$(not in LMFDB)
4.3.f_o_z_bq$3$(not in LMFDB)
4.3.aj_bq_aev_jy$6$(not in LMFDB)
4.3.ag_y_ack_ew$6$(not in LMFDB)
4.3.af_o_az_bq$6$(not in LMFDB)
4.3.ad_g_b_ag$6$(not in LMFDB)
4.3.ac_i_ak_be$6$(not in LMFDB)
4.3.a_d_ak_a$6$(not in LMFDB)
4.3.a_d_k_a$6$(not in LMFDB)
4.3.b_c_f_s$6$(not in LMFDB)
4.3.d_g_ab_ag$6$(not in LMFDB)
4.3.e_l_ba_bw$6$(not in LMFDB)
4.3.g_y_ck_ew$6$(not in LMFDB)
4.3.j_bq_ev_jy$6$(not in LMFDB)
4.3.af_k_af_ag$12$(not in LMFDB)
4.3.ac_e_ac_g$12$(not in LMFDB)
4.3.ab_ac_ab_s$12$(not in LMFDB)
4.3.b_ac_b_s$12$(not in LMFDB)
4.3.c_e_c_g$12$(not in LMFDB)
4.3.f_k_f_ag$12$(not in LMFDB)
4.3.aj_bo_ael_ja$24$(not in LMFDB)
4.3.ag_w_acg_ek$24$(not in LMFDB)
4.3.af_m_ax_bq$24$(not in LMFDB)
4.3.ad_e_ab_ag$24$(not in LMFDB)
4.3.ac_g_ao_s$24$(not in LMFDB)
4.3.ab_a_f_ag$24$(not in LMFDB)
4.3.b_a_af_ag$24$(not in LMFDB)
4.3.c_g_o_s$24$(not in LMFDB)
4.3.d_e_b_ag$24$(not in LMFDB)
4.3.f_m_x_bq$24$(not in LMFDB)
4.3.g_w_cg_ek$24$(not in LMFDB)
4.3.j_bo_el_ja$24$(not in LMFDB)