Invariants
| Base field: | $\F_{3}$ |
| Dimension: | $5$ |
| L-polynomial: | $( 1 + x + 3 x^{2} )( 1 - 5 x^{2} + 9 x^{4} )^{2}$ |
| $1 + x - 7 x^{2} - 10 x^{3} + 13 x^{4} + 43 x^{5} + 39 x^{6} - 90 x^{7} - 189 x^{8} + 81 x^{9} + 243 x^{10}$ | |
| Frobenius angles: | $\pm0.0932147493387$, $\pm0.0932147493387$, $\pm0.593214749339$, $\pm0.906785250661$, $\pm0.906785250661$ |
| Angle rank: | $1$ (numerical) |
| Cyclic group of points: | no |
| Non-cyclic primes: | $5$ |
This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.
Newton polygon
This isogeny class is ordinary.
| $p$-rank: | $5$ |
| Slopes: | $[0, 0, 0, 0, 0, 1, 1, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $125$ | $9375$ | $10952000$ | $2373046875$ | $974387046875$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $5$ | $-5$ | $20$ | $47$ | $275$ | $760$ | $2105$ | $7127$ | $19820$ | $60475$ |
Jacobians and polarizations
This isogeny class is principally polarizable, but does not contain a Jacobian.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{3^{4}}$.
Endomorphism algebra over $\F_{3}$The isogeny class factors as 1.3.b $\times$ 2.3.a_af 2 and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
|
| The base change of $A$ to $\F_{3^{4}}$ is 1.81.ah 5 and its endomorphism algebra is $\mathrm{M}_{5}($\(\Q(\sqrt{-11}) \)$)$ |
- Endomorphism algebra over $\F_{3^{2}}$
The base change of $A$ to $\F_{3^{2}}$ is 1.9.af 4 $\times$ 1.9.f. The endomorphism algebra for each factor is: - 1.9.af 4 : $\mathrm{M}_{4}($\(\Q(\sqrt{-11}) \)$)$
- 1.9.f : \(\Q(\sqrt{-11}) \).
Base change
This is a primitive isogeny class.