Properties

Label 1.512.ad
Base Field $\F_{2^{9}}$
Dimension $1$
Ordinary Yes
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{2^{9}}$
Dimension:  $1$
L-polynomial:  $1 - 3 x + 512 x^{2}$
Frobenius angles:  $\pm0.478883350743$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-2039}) \)
Galois group:  $C_2$

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 a Jacobian, and hence is principally polarizable.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 510 263160 134222310 68718970800 35184368225550 18014398756931880 9223372039575126390 4722366482751111919200 2417851639227509930031390 1237940039285435718687763800

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 510 263160 134222310 68718970800 35184368225550 18014398756931880 9223372039575126390 4722366482751111919200 2417851639227509930031390 1237940039285435718687763800

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{9}}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-2039}) \).
All geometric endomorphisms are defined over $\F_{2^{9}}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
1.512.d$2$(not in LMFDB)