Invariants
| Base field: | $\F_{347}$ |
| Dimension: | $1$ |
| L-polynomial: | $1 + 8 x + 347 x^{2}$ |
| Frobenius angles: | $\pm0.568887560362$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-331}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $12$ |
| Isomorphism classes: | 12 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $356$ | $121040$ | $41774108$ | $14498171200$ | $5030923527316$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $356$ | $121040$ | $41774108$ | $14498171200$ | $5030923527316$ | $1745729112051920$ | $605767992529349068$ | $210201493951623916800$ | $72939918400107356193476$ | $25310151684657354635301200$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which 0 are hyperelliptic):
- $y^2=x^3+64 x+64$
- $y^2=x^3+83 x+83$
- $y^2=x^3+20 x+40$
- $y^2=x^3+2 x+4$
- $y^2=x^3+199 x+199$
- $y^2=x^3+109 x+109$
- $y^2=x^3+173 x+173$
- $y^2=x^3+166 x+166$
- $y^2=x^3+44 x+88$
- $y^2=x^3+301 x+255$
- $y^2=x^3+321 x+321$
- $y^2=x^3+255 x+163$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{347}$.
Endomorphism algebra over $\F_{347}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-331}) \). |
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 |
|---|---|---|
| 1.347.ai | $2$ | (not in LMFDB) |