Invariants
| Base field: | $\F_{3}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 2 x + 4 x^{2} + 6 x^{3} + 9 x^{4}$ |
| Frobenius angles: | $\pm0.432222199768$, $\pm0.789232625405$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-8 -2 \sqrt{3}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $2$ |
| Cyclic group of points: | yes |
This isogeny class is simple and 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $22$ | $132$ | $814$ | $6864$ | $46222$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $6$ | $14$ | $30$ | $86$ | $186$ | $782$ | $2274$ | $6494$ | $19686$ | $58334$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which all are hyperelliptic):
- $y^2=x^6+2 x^5+x^4+x+2$
- $y^2=x^5+x^2+2 x$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{3}$.
Endomorphism algebra over $\F_{3}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-8 -2 \sqrt{3}})\). |
Base change
This is a primitive isogeny class.
Twists
Below is a list of all twists of this isogeny class.
| Twist | Extension degree | Common base change |
|---|---|---|
| 2.3.ac_e | $2$ | 2.9.e_k |