Properties

Label 1.8.a
Base field $\F_{2^{3}}$
Dimension $1$
$p$-rank $0$
Ordinary no
Supersingular yes
Simple yes
Geometrically simple yes
Primitive no
Principally polarizable yes
Contains a Jacobian yes

Learn more about

Invariants

Base field:  $\F_{2^{3}}$
Dimension:  $1$
L-polynomial:  $1 + 8 x^{2}$
Frobenius angles:  $\pm0.5$
Angle rank:  $0$ (numerical)
Number field:  \(\Q(\sqrt{-2}) \)
Galois group:  $C_2$
Jacobians:  1

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 the Jacobians of 1 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $9$ $81$ $513$ $3969$ $32769$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $9$ $81$ $513$ $3969$ $32769$ $263169$ $2097153$ $16769025$ $134217729$ $1073807361$

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{3}}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-2}) \).
Endomorphism algebra over $\overline{\F}_{2^{3}}$
The base change of $A$ to $\F_{2^{6}}$ is the simple isogeny class 1.64.q and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
All geometric endomorphisms are defined over $\F_{2^{6}}$.

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{3}}$.

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

Twists

Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change
1.8.ae$8$(not in LMFDB)
1.8.e$8$(not in LMFDB)