Properties

Label 4.3.aj_bm_ady_ht
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:  $( 1 - 3 x + 3 x^{2} )^{2}( 1 - 3 x + 5 x^{2} - 9 x^{3} + 9 x^{4} )$
Frobenius angles:  $\pm0.0975263560046$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.527857038681$
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
$A(\F_{q^r})$ 3 3969 430416 40916421 4586243568 355330789632 24599121127689 1899813952225917 153645866448077808 12221700189956054784

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -5 5 19 77 310 899 2347 6725 20143 59360

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ad 2 $\times$ 2.3.ad_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.cj_ddt. 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_c_a_d$2$(not in LMFDB)
4.3.ad_c_g_ap$2$(not in LMFDB)
4.3.d_c_ag_ap$2$(not in LMFDB)
4.3.d_c_a_d$2$(not in LMFDB)
4.3.j_bm_dy_ht$2$(not in LMFDB)
4.3.ag_u_abz_dy$3$(not in LMFDB)
4.3.ad_c_a_d$3$(not in LMFDB)
4.3.ad_l_abb_bw$3$(not in LMFDB)
4.3.a_c_ad_ag$3$(not in LMFDB)
4.3.d_c_ag_ap$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
4.3.ad_c_a_d$2$(not in LMFDB)
4.3.ad_c_g_ap$2$(not in LMFDB)
4.3.d_c_ag_ap$2$(not in LMFDB)
4.3.d_c_a_d$2$(not in LMFDB)
4.3.j_bm_dy_ht$2$(not in LMFDB)
4.3.ag_u_abz_dy$3$(not in LMFDB)
4.3.ad_c_a_d$3$(not in LMFDB)
4.3.ad_l_abb_bw$3$(not in LMFDB)
4.3.a_c_ad_ag$3$(not in LMFDB)
4.3.d_c_ag_ap$3$(not in LMFDB)
4.3.ad_i_as_bh$4$(not in LMFDB)
4.3.d_i_s_bh$4$(not in LMFDB)
4.3.a_c_d_ag$6$(not in LMFDB)
4.3.d_l_bb_bw$6$(not in LMFDB)
4.3.g_u_bz_dy$6$(not in LMFDB)
4.3.ad_ab_j_am$12$(not in LMFDB)
4.3.d_ab_aj_am$12$(not in LMFDB)
4.3.ad_f_aj_s$24$(not in LMFDB)
4.3.d_f_j_s$24$(not in LMFDB)