Invariants
| Base field: | $\F_{97}$ |
| Dimension: | $1$ |
| L-polynomial: | $1 + 10 x + 97 x^{2}$ |
| Frobenius angles: | $\pm0.669494215923$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-2}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $9$ |
| Isomorphism classes: | 9 |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2, 3$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $108$ | $9504$ | $910764$ | $88539264$ | $8587425708$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $108$ | $9504$ | $910764$ | $88539264$ | $8587425708$ | $832970182176$ | $80798294417004$ | $7837433671795200$ | $760231056916310508$ | $73742412699365804064$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 9 curves (of which 0 are hyperelliptic):
- $y^2=x^3+21 x+8$
- $y^2=x^3+28 x+43$
- $y^2=x^3+60 x+9$
- $y^2=x^3+24 x+23$
- $y^2=x^3+75 x+75$
- $y^2=x^3+61 x+14$
- $y^2=x^3+10 x+10$
- $y^2=x^3+29 x+48$
- $y^2=x^3+80 x+12$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{97}$.
Endomorphism algebra over $\F_{97}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-2}) \). |
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 |
|---|---|---|
| 1.97.ak | $2$ | (not in LMFDB) |