Invariants
| Base field: | $\F_{41}$ |
| Dimension: | $1$ |
| L-polynomial: | $1 + 7 x + 41 x^{2}$ |
| Frobenius angles: | $\pm0.684081270891$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-115}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $2$ |
| Isomorphism classes: | 2 |
| Cyclic group of points: | yes |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $49$ | $1715$ | $68404$ | $2828035$ | $115861529$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $49$ | $1715$ | $68404$ | $2828035$ | $115861529$ | $4749973760$ | $194754968849$ | $7984925714115$ | $327381902505364$ | $13422659513487875$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which 0 are hyperelliptic):
- $y^2=x^3+28 x+28$
- $y^2=x^3+9 x+27$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{41}$.
Endomorphism algebra over $\F_{41}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-115}) \). |
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 |
|---|---|---|
| 1.41.ah | $2$ | (not in LMFDB) |