Invariants
| Base field: | $\F_{41}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 4 x - 25 x^{2} + 164 x^{3} + 1681 x^{4}$ |
| Frobenius angles: | $\pm0.267782001470$, $\pm0.934448668136$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}, \sqrt{-37})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $4$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1825$ | $2717425$ | $4809422500$ | $7987732198825$ | $13425153346470625$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $46$ | $1616$ | $69778$ | $2826756$ | $115877726$ | $4750013558$ | $194753572766$ | $7984920565636$ | $327381914210338$ | $13422659541722576$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which all are hyperelliptic):
- $y^2=34 x^6+25 x^5+27 x^4+17 x^3+37 x^2+11 x+11$
- $y^2=24 x^6+31 x^5+2 x^4+38 x^3+17 x^2+29 x+30$
- $y^2=10 x^6+36 x^5+38 x^4+29 x^3+17 x^2+7 x+21$
- $y^2=7 x^6+22 x^5+27 x^4+9 x^3+2 x^2+2 x+26$
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 \(\Q(\sqrt{-3}, \sqrt{-37})\). |
| The base change of $A$ to $\F_{41^{3}}$ is 1.68921.qm 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-37}) \)$)$ |
Base change
This is a primitive isogeny class.