Properties

Label 1.27.j
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:  $1 + 9 x + 27 x^{2}$
Frobenius angles:  $\pm0.833333333333$
Angle rank:  $0$ (numerical)
Number field:  \(\Q(\sqrt{-3}) \)
Galois group:  $C_2$
Jacobians:  1

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 1 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 37 703 19684 532171 14342347 387459856 10460176057 282430067923 7625597484988 205891117745743

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 37 703 19684 532171 14342347 387459856 10460176057 282430067923 7625597484988 205891117745743

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 $\F_{3^{18}}$ is the simple isogeny class 1.387420489.cggc and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
All geometric endomorphisms are defined over $\F_{3^{18}}$.
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
1.27.aj$2$1.729.abb
1.27.aj$3$(not in LMFDB)
1.27.a$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
1.27.aj$2$1.729.abb
1.27.aj$3$(not in LMFDB)
1.27.a$3$(not in LMFDB)
1.27.a$6$(not in LMFDB)