Properties

Label 4.3.ah_z_ack_eq
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:  $( 1 - 3 x + 3 x^{2} )( 1 - 4 x + 10 x^{2} - 20 x^{3} + 30 x^{4} - 36 x^{5} + 27 x^{6} )$
Frobenius angles:  $\pm0.0844416807585$, $\pm0.166666666667$, $\pm0.360432408976$, $\pm0.575465777728$
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
$A(\F_{q^r})$ 8 7168 468608 40628224 3823530328 288872464384 23100875366584 1960516736761856 155090381688685184 12092309005923908608

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -3 11 24 75 267 746 2209 6931 20328 58731

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ad $\times$ 3.3.ae_k_au 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 $\times$ 3.729.abm_cpz_adepg. 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.ab_b_ac_a$2$(not in LMFDB)
4.3.b_b_c_a$2$(not in LMFDB)
4.3.h_z_ck_eq$2$(not in LMFDB)
4.3.ae_n_abg_ci$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
4.3.ab_b_ac_a$2$(not in LMFDB)
4.3.b_b_c_a$2$(not in LMFDB)
4.3.h_z_ck_eq$2$(not in LMFDB)
4.3.ae_n_abg_ci$3$(not in LMFDB)
4.3.e_n_bg_ci$6$(not in LMFDB)