Invariants
| Base field: | $\F_{2^{4}}$ |
| Dimension: | $3$ |
| L-polynomial: | $( 1 - 4 x )^{6}$ |
| $1 - 24 x + 240 x^{2} - 1280 x^{3} + 3840 x^{4} - 6144 x^{5} + 4096 x^{6}$ | |
| Frobenius angles: | $0$, $0$, $0$, $0$, $0$, $0$ |
| 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: | $[1/2, 1/2, 1/2, 1/2, 1/2, 1/2]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $729$ | $11390625$ | $62523502209$ | $274941996890625$ | $1146182576381093889$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $-7$ | $161$ | $3713$ | $64001$ | $1042433$ | $16752641$ | $268337153$ | $4294574081$ | $68717903873$ | $1099505336321$ |
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 $\F_{2^{4}}$.
Endomorphism algebra over $\F_{2^{4}}$| The isogeny class factors as 1.16.ai 3 and its endomorphism algebra is $\mathrm{M}_{3}(B)$, 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 $\F_{2^{4}}$.
| Subfield | Primitive Model |
| $\F_{2}$ | 3.2.a_ac_a |
| $\F_{2}$ | 3.2.a_g_a |