Invariants
| Base field: | $\F_{41}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 6 x - 5 x^{2} - 246 x^{3} + 1681 x^{4}$ |
| Frobenius angles: | $\pm0.0114535959467$, $\pm0.678120262613$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-2}, \sqrt{-3})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $7$ |
| Isomorphism classes: | 20 |
| Cyclic group of points: | no |
| Non-cyclic primes: | $5$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1425$ | $2748825$ | $4678560000$ | $7981403056425$ | $13421045974760625$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $36$ | $1636$ | $67878$ | $2824516$ | $115842276$ | $4749834958$ | $194754036996$ | $7984921130116$ | $327381865781238$ | $13422659272373476$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 7 curves (of which all are hyperelliptic):
- $y^2=14 x^6+35 x^5+28 x^4+21 x^3+22 x^2+8 x+14$
- $y^2=37 x^6+14 x^5+20 x^4+12 x^3+13 x^2+3 x+37$
- $y^2=6 x^6+10 x^5+24 x^4+17 x^3+23 x^2+26 x+6$
- $y^2=12 x^6+39 x^5+21 x^4+8 x^3+6 x^2+33 x+12$
- $y^2=6 x^6+6 x^5+8 x^4+17 x^3+5 x^2+40 x+6$
- $y^2=38 x^6+22 x^5+25 x^4+33 x^3+34 x^2+x+38$
- $y^2=26 x^6+36 x^5+29 x^4+15 x^3+34 x^2+38 x+26$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{41^{3}}$.
Endomorphism algebra over $\F_{41}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-2}, \sqrt{-3})\). |
| The base change of $A$ to $\F_{41^{3}}$ is 1.68921.auc 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$ |
Base change
This is a primitive isogeny class.