Invariants
| Base field: | $\F_{2^{2}}$ |
| Dimension: | $4$ |
| L-polynomial: | $( 1 - 3 x + 4 x^{2} )^{4}$ |
| $1 - 12 x + 70 x^{2} - 252 x^{3} + 609 x^{4} - 1008 x^{5} + 1120 x^{6} - 768 x^{7} + 256 x^{8}$ | |
| Frobenius angles: | $\pm0.230053456163$, $\pm0.230053456163$, $\pm0.230053456163$, $\pm0.230053456163$ |
| Angle rank: | $1$ (numerical) |
This isogeny class is not simple, not primitive, ordinary, and not supersingular. It is principally polarizable.
Newton polygon
This isogeny class is ordinary.
| $p$-rank: | $4$ |
| Slopes: | $[0, 0, 0, 0, 1, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $16$ | $65536$ | $29986576$ | $6879707136$ | $1370594684176$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $-7$ | $13$ | $101$ | $381$ | $1253$ | $4285$ | $16037$ | $63741$ | $258149$ | $1043773$ |
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.ad 4 and its endomorphism algebra is $\mathrm{M}_{4}($\(\Q(\sqrt{-7}) \)$)$ |
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_ag_a_r |