Invariants
| Base field: | $\F_{2}$ |
| Dimension: | $4$ |
| L-polynomial: | $1 - 4 x + 6 x^{2} - 4 x^{3} + 2 x^{4} - 8 x^{5} + 24 x^{6} - 32 x^{7} + 16 x^{8}$ |
| Frobenius angles: | $\pm0.0377785699724$, $\pm0.148391828106$, $\pm0.398391828106$, $\pm0.787778569972$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 8.0.18939904.2 |
| Galois group: | $D_4\times C_2$ |
| Jacobians: | $0$ |
| Isomorphism classes: | 2 |
This isogeny class is simple and geometrically simple, primitive, not ordinary, and not supersingular.
Newton polygon
| $p$-rank: | $0$ |
| Slopes: | $[1/4, 1/4, 1/4, 1/4, 3/4, 3/4, 3/4, 3/4]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1$ | $97$ | $2689$ | $67609$ | $457601$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $-1$ | $1$ | $5$ | $17$ | $9$ | $73$ | $153$ | $273$ | $545$ | $881$ |
Jacobians and polarizations
This isogeny class does not contain a Jacobian, and it is unknown whether it is principally polarizable.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{8}}$.
Endomorphism algebra over $\F_{2}$| The endomorphism algebra of this simple isogeny class is 8.0.18939904.2. |
| The base change of $A$ to $\F_{2^{8}}$ is the simple isogeny class 4.256.q_ds_glk_jitk and its endomorphism algebra is the quaternion algebra over 4.0.1088.2 with the following ramification data at primes above $2$, and unramified at all archimedean places: | ||||||
|
- Endomorphism algebra over $\F_{2^{2}}$
The base change of $A$ to $\F_{2^{2}}$ is the simple isogeny class 4.4.ae_i_ai_e and its endomorphism algebra is 8.0.18939904.2. - Endomorphism algebra over $\F_{2^{4}}$
The base change of $A$ to $\F_{2^{4}}$ is the simple isogeny class 4.16.a_i_a_q and its endomorphism algebra is 8.0.18939904.2.
Base change
This is a primitive isogeny class.