Invariants
| Base field: | $\F_{2}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 2 x^{2} + 4 x^{4}$ |
| Frobenius angles: | $\pm0.333333333333$, $\pm0.666666666667$ |
| Angle rank: | $0$ (numerical) |
| Number field: | \(\Q(\sqrt{2}, \sqrt{-3})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $1$ |
| Isomorphism classes: | 2 |
This isogeny class is simple but not geometrically simple, 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $7$ | $49$ | $49$ | $441$ | $1057$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $3$ | $9$ | $9$ | $25$ | $33$ | $33$ | $129$ | $289$ | $513$ | $1089$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:
- $y^2+y=x^5+x^4+x^3$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{6}}$.
Endomorphism algebra over $\F_{2}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{2}, \sqrt{-3})\). |
| The base change of $A$ to $\F_{2^{6}}$ is 1.64.aq 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$. |
- Endomorphism algebra over $\F_{2^{2}}$
The base change of $A$ to $\F_{2^{2}}$ is 1.4.c 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$ - Endomorphism algebra over $\F_{2^{3}}$
The base change of $A$ to $\F_{2^{3}}$ is the simple isogeny class 2.8.a_aq and its endomorphism algebra is the quaternion algebra over \(\Q(\sqrt{2}) \) ramified at both real infinite places.
Base change
This is a primitive isogeny class.