Invariants
| Base field: | $\F_{19}$ |
| Dimension: | $3$ |
| L-polynomial: | $( 1 - 3 x + 19 x^{2} )^{3}$ |
| $1 - 9 x + 84 x^{2} - 369 x^{3} + 1596 x^{4} - 3249 x^{5} + 6859 x^{6}$ | |
| Frobenius angles: | $\pm0.388176076177$, $\pm0.388176076177$, $\pm0.388176076177$ |
| Angle rank: | $1$ (numerical) |
| Cyclic group of points: | no |
| Non-cyclic primes: | $17$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $4913$ | $59776471$ | $343588336064$ | $2207308179875427$ | $15124326247672531343$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $11$ | $449$ | $7292$ | $129965$ | $2466821$ | $47024828$ | $893984879$ | $16984302485$ | $322687766468$ | $6131052414449$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains no Jacobian of a hyperelliptic curve, but it is unknown whether it contains a Jacobian of a non-hyperelliptic curve.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{19}$.
Endomorphism algebra over $\F_{19}$| The isogeny class factors as 1.19.ad 3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-67}) \)$)$ |
Base change
This is a primitive isogeny class.