Properties

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

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Invariants

Base field:  $\F_{3^{3}}$
Dimension:  $1$
Weil polynomial:  $1 - 7 x + 27 x^{2}$
Frobenius angles:  $\pm0.264757707515$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-59}) \)
Galois group:  $C_2$
Jacobians:  3

This isogeny class is simple and geometrically simple.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $1$
Slopes:  $[0, 1]$

Point counts

This isogeny class contains the Jacobians of 3 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 21 735 19908 532875 14352891 387409680 10460169993 282428545875 7625595497436 205891144928175

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 21 735 19908 532875 14352891 387409680 10460169993 282428545875 7625595497436 205891144928175

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{3}}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-59}) \).
All geometric endomorphisms are defined over $\F_{3^{3}}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
1.27.h$2$1.729.f