Invariants
| Base field: | $\F_{331}$ |
| Dimension: | $1$ |
| L-polynomial: | $1 - 32 x + 331 x^{2}$ |
| Frobenius angles: | $\pm0.157917618260$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $10$ |
| Isomorphism classes: | 10 |
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: | $1$ |
| Slopes: | $[0, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $300$ | $109200$ | $36263700$ | $12003700800$ | $3973198957500$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $300$ | $109200$ | $36263700$ | $12003700800$ | $3973198957500$ | $1315127884870800$ | $435307307458577700$ | $144086718372002515200$ | $47692703775861198420300$ | $15786284949772700784330000$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which 0 are hyperelliptic):
- $y^2=x^3+205 x+79$
- $y^2=x^3+106 x+106$
- $y^2=x^3+13 x+13$
- $y^2=x^3+147 x+294$
- $y^2=x^3+170 x+170$
- $y^2=x^3+68 x+136$
- $y^2=x^3+25 x+50$
- $y^2=x^3+317 x+317$
- $y^2=x^3+1$
- $y^2=x^3+270 x+270$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{331}$.
Endomorphism algebra over $\F_{331}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}) \). |
Base change
This is a primitive isogeny class.