Invariants
| Base field: | $\F_{2^{4}}$ |
| Dimension: | $1$ |
| L-polynomial: | $1 + 4 x + 16 x^{2}$ |
| Frobenius angles: | $\pm0.666666666667$ |
| Angle rank: | $0$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $2$ |
| Isomorphism classes: | 2 |
| Cyclic group of points: | yes |
This isogeny class is simple and geometrically simple, not 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
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $21$ | $273$ | $3969$ | $65793$ | $1049601$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $21$ | $273$ | $3969$ | $65793$ | $1049601$ | $16769025$ | $268451841$ | $4295032833$ | $68718952449$ | $1099512676353$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which 0 are hyperelliptic):
- $y^2+a^2 y=x^3$
- $y^2+a y=x^3$
where $a$ is a root of the Conway polynomial.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{12}}$.
Endomorphism algebra over $\F_{2^{4}}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}) \). |
| The base change of $A$ to $\F_{2^{12}}$ is the simple isogeny class 1.4096.aey and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$. |
Base change
This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{4}}$.
| Subfield | Primitive Model |
| $\F_{2^{2}}$ | 1.4.ac |
| $\F_{2^{2}}$ | 1.4.c |