Properties

Label 1.27.a
Base Field $\F_{3^{3}}$
Dimension $1$
Ordinary No
$p$-rank $0$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{3^{3}}$
Dimension:  $1$
Weil polynomial:  $1 + 27 x^{2}$
Frobenius angles:  $\pm0.5$
Angle rank:  $0$ (numerical)
Number field:  \(\Q(\sqrt{-3}) \)
Galois group:  $C_2$
Jacobians:  2

This isogeny class is simple and geometrically simple.

Newton polygon

This isogeny class is supersingular.

$p$-rank:  $0$
Slopes:  $[1/2, 1/2]$

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
$A(\F_{q^r})$ 28 784 19684 529984 14348908 387459856 10460353204 282428473600 7625597484988 205891160792464

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 28 784 19684 529984 14348908 387459856 10460353204 282428473600 7625597484988 205891160792464

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{3}}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}) \).
Endomorphism algebra over $\overline{\F}_{3^{3}}$
The base change of $A$ to $\F_{3^{6}}$ is the simple isogeny class 1.729.cc and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
All geometric endomorphisms are defined over $\F_{3^{6}}$.

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{3^{3}}$.

SubfieldPrimitive Model
$\F_{3}$1.3.ad
$\F_{3}$1.3.a
$\F_{3}$1.3.d

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
1.27.aj$3$(not in LMFDB)
1.27.j$3$(not in LMFDB)
1.27.j$6$(not in LMFDB)