Invariants
| Base field: | $\F_{3}$ |
| Dimension: | $3$ |
| L-polynomial: | $( 1 - 3 x + 3 x^{2} )( 1 - 2 x + 3 x^{2} )^{2}$ |
| $1 - 7 x + 25 x^{2} - 54 x^{3} + 75 x^{4} - 63 x^{5} + 27 x^{6}$ | |
| Frobenius angles: | $\pm0.166666666667$, $\pm0.304086723985$, $\pm0.304086723985$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $0$ |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.
Newton polygon
| $p$-rank: | $2$ |
| Slopes: | $[0, 0, 1/2, 1/2, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $4$ | $1008$ | $40432$ | $838656$ | $15870844$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $-3$ | $11$ | $48$ | $119$ | $267$ | $692$ | $2097$ | $6575$ | $20064$ | $59771$ |
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 $\F_{3^{6}}$.
Endomorphism algebra over $\F_{3}$The isogeny class factors as 1.3.ad $\times$ 1.3.ac 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 $\F_{3^{6}}$ is 1.729.abu 2 $\times$ 1.729.cc. The endomorphism algebra for each factor is:
|
- Endomorphism algebra over $\F_{3^{2}}$
The base change of $A$ to $\F_{3^{2}}$ is 1.9.ad $\times$ 1.9.c 2 . The endomorphism algebra for each factor is: - 1.9.ad : \(\Q(\sqrt{-3}) \).
- 1.9.c 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$
- Endomorphism algebra over $\F_{3^{3}}$
The base change of $A$ to $\F_{3^{3}}$ is 1.27.a $\times$ 1.27.k 2 . The endomorphism algebra for each factor is: - 1.27.a : \(\Q(\sqrt{-3}) \).
- 1.27.k 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$
Base change
This is a primitive isogeny class.