Invariants
| Base field: | $\F_{3}$ |
| Dimension: | $5$ |
| L-polynomial: | $1 + x^{2} + x^{3} - 4 x^{4} - 10 x^{5} - 12 x^{6} + 9 x^{7} + 27 x^{8} + 243 x^{10}$ |
| Frobenius angles: | $\pm0.0782256227849$, $\pm0.376464785396$, $\pm0.447534816427$, $\pm0.688415667813$, $\pm0.903611709235$ |
| Angle rank: | $5$ (numerical) |
| Number field: | 10.0.9152976181568663.1 |
| Galois group: | $C_2 \wr S_5$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
This isogeny class is simple and geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $256$ | $65536$ | $15391744$ | $2861563904$ | $680971636736$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $4$ | $12$ | $31$ | $64$ | $189$ | $681$ | $2356$ | $6864$ | $19183$ | $59967$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 2 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:
- $y^2=2 x^{12}+2 x^{11}+x^9+x^7+2 x^5+2 x$
- $y^2=2 x^{12}+2 x^{11}+x^{10}+2 x^7+x^4+2 x^3+x^2+x$
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 10.0.9152976181568663.1. |
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.a_b_ab_ae_k | $2$ | (not in LMFDB) |