Invariants
| Base field: | $\F_{41}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 37 x^{2} + 1681 x^{4}$ |
| Frobenius angles: | $\pm0.175494573493$, $\pm0.824505426507$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-5}, \sqrt{119})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $26$ |
| Isomorphism classes: | 72 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1645$ | $2706025$ | $4750240180$ | $7996198340025$ | $13422659143781125$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $42$ | $1608$ | $68922$ | $2829748$ | $115856202$ | $4750376118$ | $194754273882$ | $7984928588068$ | $327381934393962$ | $13422658977409848$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 26 curves (of which all are hyperelliptic):
- $y^2=31 x^6+28 x^5+4 x^4+32 x^3+22 x^2+27 x+7$
- $y^2=21 x^6+13 x^5+21 x^4+26 x^3+31 x^2+7 x+24$
- $y^2=3 x^6+37 x^5+3 x^4+33 x^3+22 x^2+x+21$
- $y^2=33 x^6+27 x^5+24 x^4+20 x^3+22 x^2+15 x+4$
- $y^2=34 x^6+39 x^5+21 x^4+38 x^3+9 x^2+8 x+24$
- $y^2=7 x^6+40 x^5+38 x^4+12 x^3+36 x^2+2 x+33$
- $y^2=x^6+35 x^5+23 x^4+31 x^3+11 x^2+12 x+34$
- $y^2=6 x^6+15 x^5+13 x+37$
- $y^2=2 x^6+36 x^5+33 x^4+3 x^3+6 x^2+10 x+3$
- $y^2=10 x^6+28 x^4+13 x^3+16 x^2+29$
- $y^2=19 x^6+7 x^5+34 x^4+5 x^3+40 x^2+6 x+4$
- $y^2=4 x^6+x^5+34 x^4+30 x^3+32 x^2+19 x+18$
- $y^2=24 x^6+6 x^5+40 x^4+16 x^3+28 x^2+32 x+26$
- $y^2=7 x^6+4 x^5+25 x^4+8 x^3+6 x^2+24 x+11$
- $y^2=x^6+24 x^5+27 x^4+7 x^3+36 x^2+21 x+25$
- $y^2=2 x^6+9 x^5+6 x^4+9 x^3+21 x^2+18 x+14$
- $y^2=10 x^6+22 x^5+40 x^4+5 x^3+11 x^2+38 x+15$
- $y^2=21 x^6+9 x^5+12 x^4+x^3+31 x^2+37 x+26$
- $y^2=18 x^6+34 x^5+x^4+34 x^3+14 x^2+22 x+28$
- $y^2=36 x^6+39 x^5+30 x^4+6 x^3+x^2+21 x+34$
- $y^2=11 x^6+29 x^5+16 x^4+36 x^3+6 x^2+3 x+40$
- $y^2=40 x^6+21 x^5+37 x^4+39 x^3+24 x^2+30 x+4$
- $y^2=35 x^6+3 x^5+17 x^4+29 x^3+21 x^2+16 x+24$
- $y^2=15 x^6+28 x^5+20 x^3+6 x+5$
- $y^2=6 x^6+34 x^5+16 x^4+9 x^3+16 x^2+16 x+10$
- $y^2=36 x^6+40 x^5+14 x^4+13 x^3+14 x^2+14 x+19$
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 \(\Q(\sqrt{-5}, \sqrt{119})\). |
| The base change of $A$ to $\F_{41^{2}}$ is 1.1681.abl 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-595}) \)$)$ |
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_bl | $4$ | (not in LMFDB) |