Properties

Label 4.3.ah_t_ay_y
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:  $( 1 - 3 x + 3 x^{2} )^{2}( 1 - x - 2 x^{2} - 3 x^{3} + 9 x^{4} )$
Frobenius angles:  $\pm0.0734519173280$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.740118583995$
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 2352 313600 58828224 3833913964 318637670400 25583626756636 1867179827525376 152206785136134400 12155794379587396272

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -3 -1 12 107 267 818 2433 6611 19956 59039

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ad 2 $\times$ 2.3.ab_ac 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.ak 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.af_h_m_abw$2$(not in LMFDB)
4.3.ab_af_a_y$2$(not in LMFDB)
4.3.b_af_a_y$2$(not in LMFDB)
4.3.f_h_am_abw$2$(not in LMFDB)
4.3.h_t_y_y$2$(not in LMFDB)
4.3.ae_h_am_y$3$(not in LMFDB)
4.3.ae_k_ay_bz$3$(not in LMFDB)
4.3.ab_af_a_y$3$(not in LMFDB)
4.3.ab_e_aj_g$3$(not in LMFDB)
4.3.ab_h_am_y$3$(not in LMFDB)
4.3.c_b_ag_am$3$(not in LMFDB)
4.3.c_e_a_ad$3$(not in LMFDB)
4.3.c_n_s_ci$3$(not in LMFDB)
4.3.f_h_am_abw$3$(not in LMFDB)
4.3.f_t_bw_ds$3$(not in LMFDB)
4.3.i_bi_ds_hn$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
4.3.af_h_m_abw$2$(not in LMFDB)
4.3.ab_af_a_y$2$(not in LMFDB)
4.3.b_af_a_y$2$(not in LMFDB)
4.3.f_h_am_abw$2$(not in LMFDB)
4.3.h_t_y_y$2$(not in LMFDB)
4.3.ae_h_am_y$3$(not in LMFDB)
4.3.ae_k_ay_bz$3$(not in LMFDB)
4.3.ab_af_a_y$3$(not in LMFDB)
4.3.ab_e_aj_g$3$(not in LMFDB)
4.3.ab_h_am_y$3$(not in LMFDB)
4.3.c_b_ag_am$3$(not in LMFDB)
4.3.c_e_a_ad$3$(not in LMFDB)
4.3.c_n_s_ci$3$(not in LMFDB)
4.3.f_h_am_abw$3$(not in LMFDB)
4.3.f_t_bw_ds$3$(not in LMFDB)
4.3.i_bi_ds_hn$3$(not in LMFDB)
4.3.ab_b_ag_m$4$(not in LMFDB)
4.3.b_b_g_m$4$(not in LMFDB)
4.3.ai_bi_ads_hn$6$(not in LMFDB)
4.3.ag_u_abw_dp$6$(not in LMFDB)
4.3.af_t_abw_ds$6$(not in LMFDB)
4.3.ad_l_ay_bw$6$(not in LMFDB)
4.3.ac_b_g_am$6$(not in LMFDB)
4.3.ac_e_a_ad$6$(not in LMFDB)
4.3.ac_n_as_ci$6$(not in LMFDB)
4.3.a_c_a_d$6$(not in LMFDB)
4.3.a_l_a_bw$6$(not in LMFDB)
4.3.b_e_j_g$6$(not in LMFDB)
4.3.b_h_m_y$6$(not in LMFDB)
4.3.d_l_y_bw$6$(not in LMFDB)
4.3.e_h_m_y$6$(not in LMFDB)
4.3.e_k_y_bz$6$(not in LMFDB)
4.3.g_u_bw_dp$6$(not in LMFDB)
4.3.ag_k_m_acf$12$(not in LMFDB)
4.3.ad_b_g_am$12$(not in LMFDB)
4.3.ac_b_g_ay$12$(not in LMFDB)
4.3.ac_k_am_bn$12$(not in LMFDB)
4.3.ab_ai_d_be$12$(not in LMFDB)
4.3.a_al_a_bw$12$(not in LMFDB)
4.3.a_ai_a_bh$12$(not in LMFDB)
4.3.a_ac_a_d$12$(not in LMFDB)
4.3.a_ab_a_am$12$(not in LMFDB)
4.3.a_b_a_am$12$(not in LMFDB)
4.3.a_i_a_bh$12$(not in LMFDB)
4.3.b_ai_ad_be$12$(not in LMFDB)
4.3.c_b_ag_ay$12$(not in LMFDB)
4.3.c_k_m_bn$12$(not in LMFDB)
4.3.d_b_ag_am$12$(not in LMFDB)
4.3.g_k_am_acf$12$(not in LMFDB)
4.3.ac_h_ag_s$24$(not in LMFDB)
4.3.ab_ac_ad_s$24$(not in LMFDB)
4.3.a_af_a_s$24$(not in LMFDB)
4.3.a_f_a_s$24$(not in LMFDB)
4.3.b_ac_d_s$24$(not in LMFDB)
4.3.c_h_g_s$24$(not in LMFDB)