Properties

Label 4.3.ai_bd_aco_eq
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:  $( 1 - 3 x + 3 x^{2} )^{2}( 1 - 2 x + 2 x^{2} - 6 x^{3} + 9 x^{4} )$
Frobenius angles:  $\pm0.116139763599$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.616139763599$
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 3920 341824 52998400 5070660404 326947819520 24919076318756 1992815309721600 151936029738601024 12058002552079778000

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -4 4 14 96 336 838 2376 7040 19922 58564

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ad 2 $\times$ 2.3.ac_c 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.sk 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.ae_f_g_ay$2$(not in LMFDB)
4.3.c_ab_a_m$2$(not in LMFDB)
4.3.i_bd_co_eq$2$(not in LMFDB)
4.3.af_o_abh_co$3$(not in LMFDB)
4.3.ac_ab_a_m$3$(not in LMFDB)
4.3.ac_i_as_be$3$(not in LMFDB)
4.3.b_c_ad_ag$3$(not in LMFDB)
4.3.e_f_ag_ay$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
4.3.ae_f_g_ay$2$(not in LMFDB)
4.3.c_ab_a_m$2$(not in LMFDB)
4.3.i_bd_co_eq$2$(not in LMFDB)
4.3.af_o_abh_co$3$(not in LMFDB)
4.3.ac_ab_a_m$3$(not in LMFDB)
4.3.ac_i_as_be$3$(not in LMFDB)
4.3.b_c_ad_ag$3$(not in LMFDB)
4.3.e_f_ag_ay$3$(not in LMFDB)
4.3.ac_f_am_y$4$(not in LMFDB)
4.3.c_f_m_y$4$(not in LMFDB)
4.3.ab_c_d_ag$6$(not in LMFDB)
4.3.b_c_ad_ag$6$(not in LMFDB)
4.3.c_i_s_be$6$(not in LMFDB)
4.3.f_o_bh_co$6$(not in LMFDB)
4.3.ag_l_g_abq$8$(not in LMFDB)
4.3.ag_t_abq_da$8$(not in LMFDB)
4.3.a_ah_a_be$8$(not in LMFDB)
4.3.a_ab_a_g$8$(not in LMFDB)
4.3.a_b_a_g$8$(not in LMFDB)
4.3.a_h_a_be$8$(not in LMFDB)
4.3.g_l_ag_abq$8$(not in LMFDB)
4.3.g_t_bq_da$8$(not in LMFDB)
4.3.ac_ae_g_g$12$(not in LMFDB)
4.3.c_ae_ag_g$12$(not in LMFDB)
4.3.ad_c_d_ag$24$(not in LMFDB)
4.3.ad_k_av_bq$24$(not in LMFDB)
4.3.ac_c_ag_s$24$(not in LMFDB)
4.3.a_ak_a_bq$24$(not in LMFDB)
4.3.a_ae_a_s$24$(not in LMFDB)
4.3.a_ac_a_ag$24$(not in LMFDB)
4.3.a_c_a_ag$24$(not in LMFDB)
4.3.a_e_a_s$24$(not in LMFDB)
4.3.a_k_a_bq$24$(not in LMFDB)
4.3.c_c_g_s$24$(not in LMFDB)
4.3.d_c_ad_ag$24$(not in LMFDB)
4.3.d_k_v_bq$24$(not in LMFDB)