Properties

Label 1.512.z
Base field $\F_{2^{9}}$
Dimension $1$
$p$-rank $1$
Ordinary yes
Supersingular no
Simple yes
Geometrically simple yes
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{2^{9}}$
Dimension:  $1$
L-polynomial:  $1 + 25 x + 512 x^{2}$
Frobenius angles:  $\pm0.686298261312$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-1423}) \)
Galois group:  $C_2$

This isogeny class is simple and geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $1$
Slopes:  $[0, 1]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $538$ $262544$ $134194954$ $68719841824$ $35184374622458$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $538$ $262544$ $134194954$ $68719841824$ $35184374622458$ $18014398259216816$ $9223372041814189034$ $4722366482873795649600$ $2417851639226615368943578$ $1237940039285444224387661264$

Jacobians and polarizations

This isogeny class contains a Jacobian, and hence is principally polarizable.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{9}}$.

Endomorphism algebra over $\F_{2^{9}}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-1423}) \).

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.az$2$(not in LMFDB)