Invariants
| Base field: | $\F_{3}$ |
| Dimension: | $5$ |
| L-polynomial: | $1 - x + 5 x^{2} - 2 x^{3} + 23 x^{4} - 19 x^{5} + 69 x^{6} - 18 x^{7} + 135 x^{8} - 81 x^{9} + 243 x^{10}$ |
| Frobenius angles: | $\pm0.228687203318$, $\pm0.340855233885$, $\pm0.389143178301$, $\pm0.691644711089$, $\pm0.755159265438$ |
| Angle rank: | $5$ (numerical) |
| Number field: | 10.0.86427828766225152.1 |
| Galois group: | $C_2 \wr S_5$ |
| Cyclic group of points: | yes |
This isogeny class is simple and geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable.
Newton polygon
This isogeny class is ordinary.
| $p$-rank: | $5$ |
| Slopes: | $[0, 0, 0, 0, 0, 1, 1, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $355$ | $211935$ | $18004180$ | $6681250875$ | $714396091025$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $3$ | $19$ | $36$ | $135$ | $203$ | $628$ | $2327$ | $6503$ | $19692$ | $58819$ |
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 $\F_{3}$.
Endomorphism algebra over $\F_{3}$| The endomorphism algebra of this simple isogeny class is 10.0.86427828766225152.1. |
Base change
This is a primitive isogeny class.
Twists
Below is a list of all twists of this isogeny class.
| Twist | Extension degree | Common base change |
|---|---|---|
| 5.3.b_f_c_x_t | $2$ | (not in LMFDB) |