Invariants
| Base field: | $\F_{2^{8}}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 16 x )^{4}$ |
| $1 - 64 x + 1536 x^{2} - 16384 x^{3} + 65536 x^{4}$ | |
| Frobenius angles: | $0$, $0$, $0$, $0$ |
| Angle rank: | $0$ (numerical) |
This isogeny class is not 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: | $[1/2, 1/2, 1/2, 1/2]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $50625$ | $4228250625$ | $281200199450625$ | $18445618199572250625$ | $1208921207935207812890625$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $193$ | $64513$ | $16760833$ | $4294705153$ | $1099507433473$ | $281474909601793$ | $72057592964186113$ | $18446744056529682433$ | $4722366482594767306753$ | $1208925819610231128195073$ |
Jacobians and polarizations
This isogeny class contains a Jacobian, and hence is principally polarizable.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{8}}$.
Endomorphism algebra over $\F_{2^{8}}$| The isogeny class factors as 1.256.abg 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, 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 $\F_{2^{8}}$.
| Subfield | Primitive Model |
| $\F_{2}$ | 2.2.ae_i |
| $\F_{2}$ | 2.2.ac_e |
| $\F_{2}$ | 2.2.a_ae |
| $\F_{2}$ | 2.2.a_a |
| $\F_{2}$ | 2.2.a_e |
| $\F_{2}$ | 2.2.c_e |
| $\F_{2}$ | 2.2.e_i |