Properties

Label 1.32.al
Base Field $\F_{2^{5}}$
Dimension $1$
Ordinary Yes
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{2^{5}}$
Dimension:  $1$
L-polynomial:  $1 - 11 x + 32 x^{2}$
Frobenius angles:  $\pm0.0751336404065$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-7}) \)
Galois group:  $C_2$
Jacobians:  1

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 1 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 22 968 32494 1047376 33550022 1073731736 34359768542 1099512282528 35184378325558 1125899954494568

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 22 968 32494 1047376 33550022 1073731736 34359768542 1099512282528 35184378325558 1125899954494568

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{5}}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-7}) \).
All geometric endomorphisms are defined over $\F_{2^{5}}$.

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{5}}$.

SubfieldPrimitive Model
$\F_{2}$1.2.ab

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
1.32.l$2$1.1024.acf