Label 1.2.c
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

Related objects


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

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

Newton polygon

This isogeny class is supersingular.

$p$-rank:  $0$
Slopes:  $[1/2, 1/2]$

Point counts

This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $5$ $5$ $5$ $25$ $25$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $5$ $5$ $5$ $25$ $25$ $65$ $145$ $225$ $545$ $1025$

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-1}) \).
Endomorphism algebra over $\overline{\F}_{2}$
The base change of $A$ to $\F_{2^{4}}$ is the simple isogeny class 1.16.i and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
All geometric endomorphisms are defined over $\F_{2^{4}}$.
Remainder of endomorphism lattice by field

Base change

This is a primitive isogeny class.


Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change$2$1.4.a