Properties

Label 2.9.af_s
Base field $\F_{3^{2}}$
Dimension $2$
$p$-rank $1$
Ordinary No
Supersingular No
Simple No
Geometrically simple No
Primitive Yes
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{3^{2}}$
Dimension:  $2$
L-polynomial:  $( 1 - 5 x + 9 x^{2} )( 1 + 9 x^{2} )$
Frobenius angles:  $\pm0.186429498677$, $\pm0.5$
Angle rank:  $1$ (numerical)
Jacobians:  2

This isogeny class is not simple.

Newton polygon

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

Point counts

This isogeny class contains the Jacobians of 2 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 50 7500 540200 42720000 3514951250 283929120000 22888831190450 1852470606720000 150086550606213800 12157702312129687500

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 5 93 740 6513 59525 534258 4785485 43033953 387399620 3486794973

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{2}}$
The isogeny class factors as 1.9.af $\times$ 1.9.a 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 $\overline{\F}_{3^{2}}$
The base change of $A$ to $\F_{3^{4}}$ is 1.81.ah $\times$ 1.81.s. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{3^{4}}$.

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
2.9.f_s$2$2.81.l_bk
2.9.al_bw$4$(not in LMFDB)
2.9.ab_am$4$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.9.f_s$2$2.81.l_bk
2.9.al_bw$4$(not in LMFDB)
2.9.ab_am$4$(not in LMFDB)
2.9.b_am$4$(not in LMFDB)
2.9.l_bw$4$(not in LMFDB)
2.9.ai_bh$12$(not in LMFDB)
2.9.ac_d$12$(not in LMFDB)
2.9.c_d$12$(not in LMFDB)
2.9.i_bh$12$(not in LMFDB)