Invariants
| Base field: | $\F_{2^{3}}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 9 x + 35 x^{2} - 72 x^{3} + 64 x^{4}$ |
| Frobenius angles: | $\pm0.0373126015494$, $\pm0.296020731784$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}, \sqrt{5})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $1$ |
| Isomorphism classes: | 1 |
This isogeny class is simple but not geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is ordinary.
| $p$-rank: | $2$ |
| Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $19$ | $3439$ | $261364$ | $16744491$ | $1064657989$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $0$ | $54$ | $513$ | $4090$ | $32490$ | $260583$ | $2092356$ | $16769074$ | $134217729$ | $1073781414$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:
- $y^2+(x^3+(a+1)x+a+1)y=x^6+(a+1)x^5+(a+1)x^4+(a+1)x^3+(a^2+1)x^2+a^2x+a^2$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{18}}$.
Endomorphism algebra over $\F_{2^{3}}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{5})\). |
| The base change of $A$ to $\F_{2^{18}}$ is 1.262144.abeb 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$ |
- Endomorphism algebra over $\F_{2^{6}}$
The base change of $A$ to $\F_{2^{6}}$ is the simple isogeny class 2.64.al_cf and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{5})\). - Endomorphism algebra over $\F_{2^{9}}$
The base change of $A$ to $\F_{2^{9}}$ is the simple isogeny class 2.512.a_abeb and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{5})\).
Base change
This is a primitive isogeny class.