Invariants
| Base field: | $\F_{3}$ |
| Dimension: | $3$ |
| L-polynomial: | $( 1 - 3 x + 3 x^{2} )( 1 - x + 3 x^{2} - 3 x^{3} + 9 x^{4} )$ |
| $1 - 4 x + 9 x^{2} - 15 x^{3} + 27 x^{4} - 36 x^{5} + 27 x^{6}$ | |
| Frobenius angles: | $\pm0.166666666667$, $\pm0.268536328535$, $\pm0.622727850897$ |
| Angle rank: | $2$ (numerical) |
| Jacobians: | $0$ |
| Isomorphism classes: | 6 |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.
Newton polygon
| $p$-rank: | $1$ |
| Slopes: | $[0, 1/2, 1/2, 1/2, 1/2, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $9$ | $1071$ | $18900$ | $737919$ | $19160784$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $0$ | $12$ | $27$ | $108$ | $315$ | $729$ | $2142$ | $6660$ | $19521$ | $58707$ |
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 $\times$ 2.3.ab_d 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.cc $\times$ 2.729.acd_deb. 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 $\times$ 2.9.f_v. The endomorphism algebra for each factor is: - Endomorphism algebra over $\F_{3^{3}}$
The base change of $A$ to $\F_{3^{3}}$ is 1.27.a $\times$ 2.27.ab_abb. The endomorphism algebra for each factor is:
Base change
This is a primitive isogeny class.