Invariants
| Base field: | $\F_{3^{2}}$ |
| Dimension: | $1$ |
| L-polynomial: | $1 + 2 x + 9 x^{2}$ |
| Frobenius angles: | $\pm0.608173447969$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-2}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $3$ |
| Isomorphism classes: | 3 |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
This isogeny class is simple and geometrically simple, not primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is ordinary.
| $p$-rank: | $1$ |
| Slopes: | $[0, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $12$ | $96$ | $684$ | $6528$ | $59532$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $12$ | $96$ | $684$ | $6528$ | $59532$ | $530784$ | $4779948$ | $43058688$ | $387423756$ | $3486670176$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 3 curves (of which 0 are hyperelliptic):
- $y^2+x y=x^3+a^7$
- $y^2+x y=x^3+1$
- $y^2+x y=x^3+a^5$
where $a$ is a root of the Conway polynomial.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{3^{2}}$.
Endomorphism algebra over $\F_{3^{2}}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-2}) \). |
Base change
This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{3^{2}}$.
| Subfield | Primitive Model |
| $\F_{3}$ | 1.3.ac |
| $\F_{3}$ | 1.3.c |
Twists
Below is a list of all twists of this isogeny class.
| Twist | Extension degree | Common base change |
|---|---|---|
| 1.9.ac | $2$ | 1.81.o |