Invariants
Base field: | $\F_{5}$ |
Dimension: | $3$ |
L-polynomial: | $( 1 - 3 x + 5 x^{2} )^{3}$ |
$1 - 9 x + 42 x^{2} - 117 x^{3} + 210 x^{4} - 225 x^{5} + 125 x^{6}$ | |
Frobenius angles: | $\pm0.265942140215$, $\pm0.265942140215$, $\pm0.265942140215$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $27$ | $19683$ | $2985984$ | $307546875$ | $32248529487$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-3$ | $29$ | $180$ | $773$ | $3297$ | $15404$ | $76605$ | $387173$ | $1950372$ | $9774629$ |
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 $\F_{5}$.
Endomorphism algebra over $\F_{5}$The isogeny class factors as 1.5.ad 3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-11}) \)$)$ |
Base change
This is a primitive isogeny class.