Invariants
| Base field: | $\F_{41}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + x - 60 x^{2} + 41 x^{3} + 1681 x^{4}$ |
| Frobenius angles: | $\pm0.149099081992$, $\pm0.922326295034$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.1618805.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $24$ |
This isogeny class is simple and 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1664$ | $2629120$ | $4771160576$ | $7982439495680$ | $13425254947621504$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $43$ | $1561$ | $69226$ | $2824881$ | $115878603$ | $4750219918$ | $194755027363$ | $7984931978721$ | $327381930454426$ | $13422659493723801$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 24 curves (of which all are hyperelliptic):
- $y^2=21 x^6+38 x^5+3 x^4+2 x^3+26 x^2+33 x$
- $y^2=8 x^6+28 x^5+9 x^4+31 x^3+14 x^2+16 x+3$
- $y^2=11 x^6+20 x^5+15 x^4+38 x^3+23 x^2+40 x+40$
- $y^2=3 x^6+16 x^5+16 x^4+27 x^3+38 x^2+16 x$
- $y^2=34 x^6+27 x^5+25 x^4+21 x^3+20 x^2+40 x+28$
- $y^2=11 x^5+35 x^4+35 x^3+26 x^2+30 x+19$
- $y^2=16 x^6+17 x^5+38 x^4+12 x^3+10 x^2+23 x$
- $y^2=39 x^6+6 x^4+7 x^3+34 x^2+4 x+12$
- $y^2=36 x^6+28 x^5+4 x^4+18 x^3+6 x^2+8 x+20$
- $y^2=32 x^6+x^5+32 x^4+5 x^3+27 x^2+7 x$
- $y^2=20 x^6+28 x^5+15 x^4+3 x^3+20 x^2+39 x+35$
- $y^2=29 x^6+26 x^5+25 x^4+15 x^3+19 x^2+35 x+28$
- $y^2=28 x^6+13 x^5+5 x^4+14 x^3+8 x^2+38 x+3$
- $y^2=34 x^6+34 x^5+27 x^4+6 x^3+14 x^2+27 x+33$
- $y^2=9 x^6+15 x^5+37 x^4+13 x^3+24 x^2+5 x+16$
- $y^2=6 x^6+19 x^5+15 x^4+32 x^3+12 x^2+25 x+38$
- $y^2=13 x^6+9 x^5+16 x^4+25 x^3+10 x^2+7 x+26$
- $y^2=4 x^5+5 x^4+22 x^3+11 x^2+13 x+37$
- $y^2=11 x^6+39 x^5+x^4+30 x^3+22 x+19$
- $y^2=4 x^6+35 x^5+36 x^4+20 x^3+x^2+24 x+38$
- $y^2=33 x^6+10 x^5+35 x^4+26 x^3+7 x^2+33 x+9$
- $y^2=6 x^6+38 x^5+31 x^4+19 x^3+31 x^2+4 x+28$
- $y^2=34 x^6+22 x^5+6 x^4+23 x^3+4 x^2+35 x+21$
- $y^2=12 x^5+8 x^4+16 x^3+16 x^2+3$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{41}$.
Endomorphism algebra over $\F_{41}$| The endomorphism algebra of this simple isogeny class is 4.0.1618805.1. |
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.ab_aci | $2$ | (not in LMFDB) |