Invariants
| Base field: | $\F_{41}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 79 x^{2} + 1681 x^{4}$ |
| Frobenius angles: | $\pm0.0431839009187$, $\pm0.956816099081$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}, \sqrt{161})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $2$ |
| Isomorphism classes: | 4 |
| Cyclic group of points: | yes |
This isogeny class is simple but not geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is ordinary.
| $p$-rank: | $2$ |
| Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1603$ | $2569609$ | $4750009600$ | $7968668431689$ | $13422659260913203$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $42$ | $1524$ | $68922$ | $2820004$ | $115856202$ | $4749914958$ | $194754273882$ | $7984919954884$ | $327381934393962$ | $13422659211674004$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which all are hyperelliptic):
- $y^2=30 x^6+37 x^5+18 x^4+6 x^3+28 x^2+5 x+10$
- $y^2=16 x^6+22 x^5+28 x^4+33 x^3+2 x^2+34 x+22$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{41^{2}}$.
Endomorphism algebra over $\F_{41}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{161})\). |
| The base change of $A$ to $\F_{41^{2}}$ is 1.1681.adb 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-483}) \)$)$ |
Base change
This is a primitive isogeny class.