Invariants
| Base field: | $\F_{41}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 24 x^{2} + 1681 x^{4}$ |
| Frobenius angles: | $\pm0.202725979435$, $\pm0.797274020565$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-58}, \sqrt{106})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $8$ |
| Isomorphism classes: | 32 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1658$ | $2748964$ | $4750211450$ | $8000683788304$ | $13422659079289178$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $42$ | $1634$ | $68922$ | $2831334$ | $115856202$ | $4750318658$ | $194754273882$ | $7984921008574$ | $327381934393962$ | $13422658848425954$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):
- $y^2=34 x^6+9 x^5+20 x^4+36 x^3+22 x^2+5 x+15$
- $y^2=40 x^6+13 x^5+38 x^4+11 x^3+9 x^2+30 x+8$
- $y^2=14 x^6+22 x^5+36 x^4+31 x^3+26 x^2+28 x+25$
- $y^2=2 x^6+9 x^5+11 x^4+22 x^3+33 x^2+4 x+27$
- $y^2=37 x^6+11 x^5+35 x^4+36 x^3+4 x^2+4 x+23$
- $y^2=17 x^6+25 x^5+5 x^4+11 x^3+24 x^2+24 x+15$
- $y^2=26 x^6+13 x^5+39 x^4+8 x^3+14 x^2+16 x+18$
- $y^2=33 x^6+37 x^5+29 x^4+7 x^3+2 x^2+14 x+26$
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 \(\Q(\sqrt{-58}, \sqrt{106})\). |
| The base change of $A$ to $\F_{41^{2}}$ is 1.1681.ay 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1537}) \)$)$ |
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 |
|---|---|---|
| 2.41.a_y | $4$ | (not in LMFDB) |