Properties

Label 4.3.ah_y_acd_dy
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:  $( 1 - 3 x + 3 x^{2} )( 1 - 2 x + 3 x^{2} )( 1 - 2 x + 2 x^{2} - 6 x^{3} + 9 x^{4} )$
Frobenius angles:  $\pm0.116139763599$, $\pm0.166666666667$, $\pm0.304086723985$, $\pm0.616139763599$
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
$A(\F_{q^r})$ 8 6720 463904 55910400 4528043608 285245291520 23085014729848 1958316775833600 153402593732216864 12206659206053928000

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -3 9 24 101 307 738 2209 6925 20112 59289

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_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 $\times$ 1.531441.azi $\times$ 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.ad_e_al_be$2$(not in LMFDB)
4.3.ad_e_f_as$2$(not in LMFDB)
4.3.ab_a_ab_g$2$(not in LMFDB)
4.3.b_a_b_g$2$(not in LMFDB)
4.3.d_e_af_as$2$(not in LMFDB)
4.3.d_e_l_be$2$(not in LMFDB)
4.3.h_y_cd_dy$2$(not in LMFDB)
4.3.ae_m_abc_cc$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
4.3.ad_e_al_be$2$(not in LMFDB)
4.3.ad_e_f_as$2$(not in LMFDB)
4.3.ab_a_ab_g$2$(not in LMFDB)
4.3.b_a_b_g$2$(not in LMFDB)
4.3.d_e_af_as$2$(not in LMFDB)
4.3.d_e_l_be$2$(not in LMFDB)
4.3.h_y_cd_dy$2$(not in LMFDB)
4.3.ae_m_abc_cc$3$(not in LMFDB)
4.3.a_e_ai_g$6$(not in LMFDB)
4.3.a_e_i_g$6$(not in LMFDB)
4.3.e_m_bc_cc$6$(not in LMFDB)
4.3.af_i_f_abe$8$(not in LMFDB)
4.3.af_q_abj_co$8$(not in LMFDB)
4.3.ab_ae_b_s$8$(not in LMFDB)
4.3.ab_e_ah_s$8$(not in LMFDB)
4.3.b_ae_ab_s$8$(not in LMFDB)
4.3.b_e_h_s$8$(not in LMFDB)
4.3.f_i_af_abe$8$(not in LMFDB)
4.3.f_q_bj_co$8$(not in LMFDB)
4.3.ac_c_c_ag$24$(not in LMFDB)
4.3.ac_k_ao_bq$24$(not in LMFDB)
4.3.c_c_ac_ag$24$(not in LMFDB)
4.3.c_k_o_bq$24$(not in LMFDB)