Properties

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

Learn more about

Invariants

Base field:  $\F_{2^{5}}$
Dimension:  $1$
L-polynomial:  $1 - 5 x + 32 x^{2}$
Frobenius angles:  $\pm0.354289791955$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-103}) \)
Galois group:  $C_2$
Jacobians:  5

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 5 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 1064 33124 1049104 33545708 1073681336 34359715124 1099513447200 35184381929788 1125899897825864

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 28 1064 33124 1049104 33545708 1073681336 34359715124 1099513447200 35184381929788 1125899897825864

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{5}}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-103}) \).
All geometric endomorphisms are defined over $\F_{2^{5}}$.

Base change

This is a primitive isogeny class.

Twists

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