Properties

Label 1.2.a
Base field $\F_{2}$
Dimension $1$
$p$-rank $0$
Ordinary no
Supersingular yes
Simple yes
Geometrically simple yes
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

Learn more about

Invariants

Base field:  $\F_{2}$
Dimension:  $1$
L-polynomial:  $1 + 2 x^{2}$
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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $3$ $9$ $9$ $9$ $33$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $3$ $9$ $9$ $9$ $33$ $81$ $129$ $225$ $513$ $1089$

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-2}) \).
Endomorphism algebra over $\overline{\F}_{2}$
The base change of $A$ to $\F_{2^{2}}$ is the simple isogeny class 1.4.e and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
All geometric endomorphisms are defined over $\F_{2^{2}}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change
1.2.ac$8$1.256.abg
1.2.c$8$1.256.abg