Invariants
| Base field: | $\F_{97}$ |
| Dimension: | $1$ |
| L-polynomial: | $1 + 8 x + 97 x^{2}$ |
| Frobenius angles: | $\pm0.633124938748$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-1}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $9$ |
| Isomorphism classes: | 9 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $106$ | $9540$ | $910858$ | $88531200$ | $8587501066$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $106$ | $9540$ | $910858$ | $88531200$ | $8587501066$ | $832970532420$ | $80798280659818$ | $7837433767756800$ | $760231057637901226$ | $73742412680808293700$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 9 curves (of which 0 are hyperelliptic):
- $y^2=x^3+7 x$
- $y^2=x^3+40 x+6$
- $y^2=x^3+53 x+71$
- $y^2=x^3+59 x+4$
- $y^2=x^3+39 x+1$
- $y^2=x^3+83 x+83$
- $y^2=x^3+93 x+77$
- $y^2=x^3+43 x+43$
- $y^2=x^3+70 x+59$
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 \(\Q(\sqrt{-1}) \). |
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 |
|---|---|---|
| 1.97.ai | $2$ | (not in LMFDB) |
| 1.97.as | $4$ | (not in LMFDB) |
| 1.97.s | $4$ | (not in LMFDB) |