Invariants
Base field: | $\F_{3^{3}}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 10 x + 27 x^{2} )^{2}$ |
$1 - 20 x + 154 x^{2} - 540 x^{3} + 729 x^{4}$ | |
Frobenius angles: | $\pm0.0877398280459$, $\pm0.0877398280459$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $1$ |
This isogeny class is not simple, not 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $324$ | $467856$ | $380016036$ | $281731654656$ | $205849551061764$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $8$ | $638$ | $19304$ | $530126$ | $14346008$ | $387427022$ | $10460496824$ | $282430796318$ | $7625606205608$ | $205891185285278$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:
- $y^2=ax^6+(2a^2+2)x^5+ax^4+ax^2+(a^2+1)x+a$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{3^{3}}$.
Endomorphism algebra over $\F_{3^{3}}$The isogeny class factors as 1.27.ak 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$ |
Base change
This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{3^{3}}$.
Subfield | Primitive Model |
$\F_{3}$ | 2.3.ac_b |
$\F_{3}$ | 2.3.e_k |