Invariants
| Base field: | $\F_{41}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 18 x + 161 x^{2} - 738 x^{3} + 1681 x^{4}$ |
| Frobenius angles: | $\pm0.197716147647$, $\pm0.298200122822$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.4672.2 |
| Galois group: | $D_{4}$ |
| Jacobians: | $9$ |
| Isomorphism classes: | 9 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1087$ | $2825113$ | $4794978748$ | $8000270823033$ | $13425222525857407$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $24$ | $1680$ | $69570$ | $2831188$ | $115878324$ | $4750110438$ | $194753738868$ | $7984921806244$ | $327381926304978$ | $13422659310378240$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 9 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=28x^6+14x^5+12x^4+38x^3+37x^2+15x+4$
- $y^2=12x^6+37x^5+21x^4+10x^3+8x+9$
- $y^2=30x^6+3x^5+20x^4+33x^3+14x^2+25x+26$
- $y^2=14x^6+x^5+14x^4+x^3+30x^2+17x+12$
- $y^2=3x^6+8x^5+29x^4+28x^3+9x^2+32x+29$
- $y^2=30x^6+36x^5+3x^4+8x^3+19x^2+22x+23$
- $y^2=27x^6+37x^5+15x^4+12x^3+27x^2+39x+26$
- $y^2=12x^6+16x^5+22x^4+29x^3+7x^2+37x+14$
- $y^2=15x^6+9x^5+16x^4+14x^3+16x^2+8x+7$
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 4.0.4672.2. |
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.s_gf | $2$ | (not in LMFDB) |