Properties

Label 4.3.ai_bg_adh_gm
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:  $( 1 - 3 x + 3 x^{2} )( 1 - 2 x + 3 x^{2} )( 1 - 3 x + 5 x^{2} - 9 x^{3} + 9 x^{4} )$
Frobenius angles:  $\pm0.0975263560046$, $\pm0.166666666667$, $\pm0.304086723985$, $\pm0.527857038681$
Angle rank:  $3$ (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
$A(\F_{q^r})$ 6 6804 584136 43164576 4095464736 310007984832 22788608466462 1866925407215232 155128934657036088 12372374984414505984

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -4 10 29 82 281 799 2180 6610 20333 60085

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ad $\times$ 1.3.ac $\times$ 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.abu $\times$ 1.729.cc $\times$ 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.ae_i_ar_bk$2$(not in LMFDB)
4.3.ac_c_ab_a$2$(not in LMFDB)
4.3.ac_c_f_am$2$(not in LMFDB)
4.3.c_c_af_am$2$(not in LMFDB)
4.3.c_c_b_a$2$(not in LMFDB)
4.3.e_i_r_bk$2$(not in LMFDB)
4.3.i_bg_dh_gm$2$(not in LMFDB)
4.3.af_r_abr_dg$3$(not in LMFDB)
4.3.ac_c_ab_a$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
4.3.ae_i_ar_bk$2$(not in LMFDB)
4.3.ac_c_ab_a$2$(not in LMFDB)
4.3.ac_c_f_am$2$(not in LMFDB)
4.3.c_c_af_am$2$(not in LMFDB)
4.3.c_c_b_a$2$(not in LMFDB)
4.3.e_i_r_bk$2$(not in LMFDB)
4.3.i_bg_dh_gm$2$(not in LMFDB)
4.3.af_r_abr_dg$3$(not in LMFDB)
4.3.ac_c_ab_a$3$(not in LMFDB)
4.3.ab_f_al_m$6$(not in LMFDB)
4.3.b_f_l_m$6$(not in LMFDB)
4.3.f_r_br_dg$6$(not in LMFDB)