Properties

 Label 1.4.ae Base field $\F_{2^{2}}$ Dimension $1$ $p$-rank $0$ Ordinary no Supersingular yes Simple yes Geometrically simple yes Primitive yes Principally polarizable yes Contains a Jacobian yes

Invariants

 Base field: $\F_{2^{2}}$ Dimension: $1$ L-polynomial: $( 1 - 2 x )^{2}$ $1 - 4x + 4x^{2}$ Frobenius angles: $0$, $0$ Angle rank: $0$ (numerical) Number field: $$\Q$$ Galois group: Trivial 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:

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $1$ $9$ $49$ $225$ $961$

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $1$ $9$ $49$ $225$ $961$ $3969$ $16129$ $65025$ $261121$ $1046529$

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
 The endomorphism algebra of this simple isogeny class 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 are some of the twists of this isogeny class.
TwistExtension degreeCommon base change
1.4.e$2$1.16.ai
1.4.c$3$1.64.aq
1.4.a$4$1.256.abg
Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change
1.4.e$2$1.16.ai
1.4.c$3$1.64.aq
1.4.a$4$1.256.abg
1.4.ac$6$(not in LMFDB)