Invariants
| Base field: | $\F_{3^{2}}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 8 x + 32 x^{2} - 72 x^{3} + 81 x^{4}$ |
| Frobenius angles: | $\pm0.141826552031$, $\pm0.358173447969$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\zeta_{8})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $3$ |
| Isomorphism classes: | 3 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $34$ | $6596$ | $561442$ | $43507216$ | $3481591874$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $2$ | $82$ | $770$ | $6630$ | $58962$ | $531442$ | $4787442$ | $43070654$ | $387475970$ | $3486784402$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(2a+2)x^6+ax^5+(2a+2)x^4+(a+1)x^3+x^2+(a+2)x+a+2$
- $y^2=2ax^6+2ax^5+2x^4+2x^3+2ax^2+ax+a$
- $y^2=x^5+2ax$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{3^{8}}$.
Endomorphism algebra over $\F_{3^{2}}$| The endomorphism algebra of this simple isogeny class is \(\Q(\zeta_{8})\). |
| The base change of $A$ to $\F_{3^{8}}$ is 1.6561.bi 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$ |
- Endomorphism algebra over $\F_{3^{4}}$
The base change of $A$ to $\F_{3^{4}}$ is the simple isogeny class 2.81.a_bi and its endomorphism algebra is \(\Q(\zeta_{8})\).
Base change
This is a primitive isogeny class.