Properties

Label 4.3.ah_ba_aco_ez
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:  $( 1 - 3 x + 3 x^{2} )^{2}( 1 - x + 5 x^{2} - 3 x^{3} + 9 x^{4} )$
Frobenius angles:  $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.345303779071$, $\pm0.557095674046$
Angle rank:  $2$ (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
$A(\F_{q^r})$ 11 10241 681296 50191141 4407635056 314606141696 23385096417001 1928181068114301 154310471689556336 12051248646952798976

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -3 13 33 93 302 811 2237 6821 20229 58528

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ad 2 $\times$ 2.3.ab_f 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.cc 2 $\times$ 2.729.abb_ef. 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_o_abe_cf$2$(not in LMFDB)
4.3.ab_c_a_d$2$(not in LMFDB)
4.3.b_c_a_d$2$(not in LMFDB)
4.3.f_o_be_cf$2$(not in LMFDB)
4.3.h_ba_co_ez$2$(not in LMFDB)
4.3.ae_o_abh_co$3$(not in LMFDB)
4.3.ab_c_a_d$3$(not in LMFDB)
4.3.ab_l_aj_bw$3$(not in LMFDB)
4.3.c_i_p_be$3$(not in LMFDB)
4.3.f_o_be_cf$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
4.3.af_o_abe_cf$2$(not in LMFDB)
4.3.ab_c_a_d$2$(not in LMFDB)
4.3.b_c_a_d$2$(not in LMFDB)
4.3.f_o_be_cf$2$(not in LMFDB)
4.3.h_ba_co_ez$2$(not in LMFDB)
4.3.ae_o_abh_co$3$(not in LMFDB)
4.3.ab_c_a_d$3$(not in LMFDB)
4.3.ab_l_aj_bw$3$(not in LMFDB)
4.3.c_i_p_be$3$(not in LMFDB)
4.3.f_o_be_cf$3$(not in LMFDB)
4.3.ab_i_ag_bh$4$(not in LMFDB)
4.3.b_i_g_bh$4$(not in LMFDB)
4.3.ac_i_ap_be$6$(not in LMFDB)
4.3.b_l_j_bw$6$(not in LMFDB)
4.3.e_o_bh_co$6$(not in LMFDB)
4.3.ab_ab_d_am$12$(not in LMFDB)
4.3.b_ab_ad_am$12$(not in LMFDB)
4.3.ab_f_ad_s$24$(not in LMFDB)
4.3.b_f_d_s$24$(not in LMFDB)