Properties

Label 2.81.abg_qc
Base Field $\F_{3^{4}}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{3^{4}}$
Dimension:  $2$
L-polynomial:  $( 1 - 16 x + 81 x^{2} )^{2}$
Frobenius angles:  $\pm0.151478024726$, $\pm0.151478024726$
Angle rank:  $1$ (numerical)
Jacobians:  16

This isogeny class is not simple.

Newton polygon

This isogeny class is ordinary.

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

Point counts

This isogeny class contains the Jacobians of 16 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 4356 41835024 282209562756 1853389289816064 12158261183469270276 79767019018955247210000 523348062861561142272444036 3433684071279087378501042438144 22528399641786318946630573420927236 147808829406475270192486728160410157584

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 50 6374 531026 43055294 3486955250 282431575718 22876811244050 1853020324299134 150094635942239666 12157665458409546854

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
The isogeny class factors as 1.81.aq 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-17}) \)$)$
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
2.81.a_adq$2$(not in LMFDB)
2.81.bg_qc$2$(not in LMFDB)
2.81.q_gt$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.81.a_adq$2$(not in LMFDB)
2.81.bg_qc$2$(not in LMFDB)
2.81.q_gt$3$(not in LMFDB)
2.81.a_dq$4$(not in LMFDB)
2.81.aq_gt$6$(not in LMFDB)