Invariants
| Base field: | $\F_{41}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 19 x + 163 x^{2} - 779 x^{3} + 1681 x^{4}$ |
| Frobenius angles: | $\pm0.0648514433265$, $\pm0.331738269742$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.1123949.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $7$ |
| Isomorphism classes: | 7 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1047$ | $2767221$ | $4756563927$ | $7983324663381$ | $13420159017819312$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $23$ | $1647$ | $69017$ | $2825195$ | $115834618$ | $4749919407$ | $194753678821$ | $7984928190259$ | $327381979946747$ | $13422659540145102$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 7 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=24x^6+21x^5+13x^4+20x^3+2x^2+5x+30$
- $y^2=22x^6+13x^5+30x^4+3x^3+12x^2+5x+26$
- $y^2=40x^6+28x^5+35x^4+11x^3+16x^2+7x+25$
- $y^2=3x^6+25x^5+21x^4+39x^3+33x^2+23x+33$
- $y^2=21x^6+18x^5+5x^4+40x^3+20x^2+5x+36$
- $y^2=21x^6+23x^5+6x^4+36x^3+6x^2+8x+10$
- $y^2=25x^6+3x^5+11x^4+38x^3+6x^2+15x+17$
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 4.0.1123949.1. |
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.t_gh | $2$ | (not in LMFDB) |