Invariants
Base field: | $\F_{7}$ |
Dimension: | $3$ |
L-polynomial: | $( 1 - 4 x + 7 x^{2} )^{3}$ |
$1 - 12 x + 69 x^{2} - 232 x^{3} + 483 x^{4} - 588 x^{5} + 343 x^{6}$ | |
Frobenius angles: | $\pm0.227185525829$, $\pm0.227185525829$, $\pm0.227185525829$ |
Angle rank: | $1$ (numerical) |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $64$ | $110592$ | $48228544$ | $15550119936$ | $4951246821184$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-4$ | $44$ | $404$ | $2684$ | $17516$ | $118508$ | $822020$ | $5752700$ | $40315868$ | $282409004$ |
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 $\F_{7}$.
Endomorphism algebra over $\F_{7}$The isogeny class factors as 1.7.ae 3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-3}) \)$)$ |
Base change
This is a primitive isogeny class.