Properties

Label 3.9.an_dc_alo
Base field $\F_{3^{2}}$
Dimension $3$
$p$-rank $2$
Ordinary no
Supersingular no
Simple no
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian no

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Invariants

Base field:  $\F_{3^{2}}$
Dimension:  $3$
L-polynomial:  $( 1 - 3 x )^{2}( 1 - 7 x + 29 x^{2} - 63 x^{3} + 81 x^{4} )$
  $1 - 13 x + 80 x^{2} - 300 x^{3} + 720 x^{4} - 1053 x^{5} + 729 x^{6}$
Frobenius angles:  $0$, $0$, $\pm0.220419591014$, $\pm0.370053256546$
Angle rank:  $2$ (numerical)

This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.

Newton polygon

$p$-rank:  $2$
Slopes:  $[0, 0, 1/2, 1/2, 1, 1]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $164$ $474944$ $399138116$ $282354208000$ $204292086406144$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-3$ $73$ $753$ $6561$ $58592$ $529633$ $4779233$ $43036961$ $387361797$ $3486502108$

Jacobians and polarizations

This isogeny class is principally polarizable, but does not contain a Jacobian.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{3^{2}}$.

Endomorphism algebra over $\F_{3^{2}}$
The isogeny class factors as 1.9.ag $\times$ 2.9.ah_bd and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

TwistExtension degreeCommon base change
3.9.ab_ae_bw$2$(not in LMFDB)
3.9.b_ae_abw$2$(not in LMFDB)
3.9.n_dc_lo$2$(not in LMFDB)
3.9.ae_r_abn$3$(not in LMFDB)

Below is a list of all twists of this isogeny class.

TwistExtension degreeCommon base change
3.9.ab_ae_bw$2$(not in LMFDB)
3.9.b_ae_abw$2$(not in LMFDB)
3.9.n_dc_lo$2$(not in LMFDB)
3.9.ae_r_abn$3$(not in LMFDB)
3.9.ah_bm_aew$4$(not in LMFDB)
3.9.h_bm_ew$4$(not in LMFDB)
3.9.ak_ch_aif$6$(not in LMFDB)
3.9.e_r_bn$6$(not in LMFDB)
3.9.k_ch_if$6$(not in LMFDB)