Properties

Label 3.3.af_l_as
Base field $\F_{3}$
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:  $\F_{3}$
Dimension:  $3$
L-polynomial:  $( 1 - 3 x + 3 x^{2} )( 1 - 2 x + 2 x^{2} - 6 x^{3} + 9 x^{4} )$
Frobenius angles:  $\pm0.116139763599$, $\pm0.166666666667$, $\pm0.616139763599$
Angle rank:  $1$ (numerical)
Jacobians:  0

This isogeny class is not simple.

Newton polygon

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

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
$A(\F_{q^r})$ 4 560 12208 582400 18710924 417025280 10982404724 299987251200 7718757861136 205043660654000

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -1 7 14 87 309 784 2295 6959 19922 58807

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ad $\times$ 2.3.ac_c 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 $\F_{3^{12}}$ is 1.531441.acec $\times$ 1.531441.sk 2 . The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{3^{12}}$.
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
3.3.ab_ab_g$2$3.9.ad_h_g
3.3.f_l_s$2$3.9.ad_h_g
3.3.ac_f_am$3$(not in LMFDB)
3.3.b_ab_ag$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.3.ab_ab_g$2$3.9.ad_h_g
3.3.f_l_s$2$3.9.ad_h_g
3.3.ac_f_am$3$(not in LMFDB)
3.3.b_ab_ag$3$(not in LMFDB)
3.3.ac_f_am$6$(not in LMFDB)
3.3.c_f_m$6$(not in LMFDB)
3.3.ad_ab_m$8$(not in LMFDB)
3.3.ad_h_am$8$(not in LMFDB)
3.3.d_ab_am$8$(not in LMFDB)
3.3.d_h_m$8$(not in LMFDB)
3.3.a_ab_a$24$(not in LMFDB)
3.3.a_h_a$24$(not in LMFDB)