Properties

Label 1.512.bg
Base Field $\F_{2^{9}}$
Dimension $1$
Ordinary No
$p$-rank $0$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{2^{9}}$
Dimension:  $1$
L-polynomial:  $1 + 32 x + 512 x^{2}$
Frobenius angles:  $\pm0.750000000000$
Angle rank:  $0$ (numerical)
Number field:  \(\Q(\sqrt{-1}) \)
Galois group:  $C_2$

This isogeny class is simple and geometrically simple.

Newton polygon

This isogeny class is supersingular.

$p$-rank:  $0$
Slopes:  $[1/2, 1/2]$

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
$A(\F_{q^r})$ 545 262145 134201345 68720001025 35184363700225 18014398509481985 9223372041149743105 4722366482732206260225 2417851639231457372667905 1237940039285380274899124225

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 545 262145 134201345 68720001025 35184363700225 18014398509481985 9223372041149743105 4722366482732206260225 2417851639231457372667905 1237940039285380274899124225

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{9}}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-1}) \).
Endomorphism algebra over $\overline{\F}_{2^{9}}$
The base change of $A$ to $\F_{2^{36}}$ is the simple isogeny class 1.68719476736.bdvoy and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
All geometric endomorphisms are defined over $\F_{2^{36}}$.
Remainder of endomorphism lattice by field

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{9}}$.

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

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
1.512.abg$2$(not in LMFDB)
1.512.a$8$(not in LMFDB)