Invariants
| Base field: | $\F_{19}$ |
| Dimension: | $3$ |
| L-polynomial: | $( 1 - x + 19 x^{2} )^{3}$ |
| $1 - 3 x + 60 x^{2} - 115 x^{3} + 1140 x^{4} - 1083 x^{5} + 6859 x^{6}$ | |
| Frobenius angles: | $\pm0.463406802480$, $\pm0.463406802480$, $\pm0.463406802480$ |
| Angle rank: | $1$ (numerical) |
| Cyclic group of points: | no |
| Non-cyclic primes: | $19$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $6859$ | $63521199$ | $330799583296$ | $2180563659421875$ | $15149696376055325869$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $17$ | $473$ | $7028$ | $128381$ | $2470967$ | $47077628$ | $894001013$ | $16983089141$ | $322684767692$ | $6131072331833$ |
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.ab 3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-3}) \)$)$ |
Base change
This is a primitive isogeny class.