Invariants
| Base field: | $\F_{2^{2}}$ |
| Dimension: | $4$ |
| L-polynomial: | $( 1 - 2 x )^{8}$ |
| $1 - 16 x + 112 x^{2} - 448 x^{3} + 1120 x^{4} - 1792 x^{5} + 1792 x^{6} - 1024 x^{7} + 256 x^{8}$ | |
| Frobenius angles: | $0$, $0$, $0$, $0$, $0$, $0$, $0$, $0$ |
| Angle rank: | $0$ (numerical) |
This isogeny class is not simple, not primitive, not ordinary, and supersingular. It is principally polarizable.
Newton polygon
This isogeny class is supersingular.
| $p$-rank: | $0$ |
| Slopes: | $[1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1$ | $6561$ | $5764801$ | $2562890625$ | $852891037441$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $-11$ | $-15$ | $1$ | $129$ | $769$ | $3585$ | $15361$ | $63489$ | $258049$ | $1040385$ |
Jacobians and polarizations
This isogeny class is principally polarizable, but does not contain a Jacobian.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{2}}$.
Endomorphism algebra over $\F_{2^{2}}$| The isogeny class factors as 1.4.ae 4 and its endomorphism algebra is $\mathrm{M}_{4}(B)$, where $B$ 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 $\F_{2^{2}}$.
| Subfield | Primitive Model |
| $\F_{2}$ | 4.2.a_ai_a_y |