Invariants
| Base field: | $\F_{3}$ |
| Dimension: | $5$ |
| L-polynomial: | $1 - 2 x + 5 x^{2} - 12 x^{3} + 22 x^{4} - 39 x^{5} + 66 x^{6} - 108 x^{7} + 135 x^{8} - 162 x^{9} + 243 x^{10}$ |
| Frobenius angles: | $\pm0.0797874469393$, $\pm0.275744318184$, $\pm0.488202455651$, $\pm0.618068781307$, $\pm0.752956898175$ |
| Angle rank: | $5$ (numerical) |
| Number field: | 10.0.178768107438520610127.1 |
| Galois group: | $C_2 \wr S_5$ |
| Cyclic group of points: | yes |
This isogeny class is simple and geometrically simple, primitive, not ordinary, and not supersingular. It is principally polarizable.
Newton polygon
| $p$-rank: | $4$ |
| Slopes: | $[0, 0, 0, 0, 1/2, 1/2, 1, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $149$ | $118455$ | $8554388$ | $3698757375$ | $863466620659$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $2$ | $16$ | $14$ | $88$ | $247$ | $724$ | $2151$ | $6344$ | $19913$ | $59791$ |
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 10.0.178768107438520610127.1. |
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.c_f_m_w_bn | $2$ | (not in LMFDB) |