Invariants
Base field: | $\F_{3}$ |
Dimension: | $3$ |
L-polynomial: | $( 1 - 2 x + 3 x^{2} )^{3}$ |
$1 - 6 x + 21 x^{2} - 44 x^{3} + 63 x^{4} - 54 x^{5} + 27 x^{6}$ | |
Frobenius angles: | $\pm0.304086723985$, $\pm0.304086723985$, $\pm0.304086723985$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $0$ |
This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.
Newton polygon
This isogeny class is ordinary.
$p$-rank: | $3$ |
Slopes: | $[0, 0, 0, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $8$ | $1728$ | $54872$ | $884736$ | $14172488$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-2$ | $16$ | $58$ | $124$ | $238$ | $592$ | $1930$ | $6460$ | $20254$ | $60496$ |
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 $\F_{3}$.
Endomorphism algebra over $\F_{3}$The isogeny class factors as 1.3.ac 3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-2}) \)$)$ |
Base change
This is a primitive isogeny class.