Invariants
| Base field: | $\F_{41}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 8 x + 23 x^{2} - 328 x^{3} + 1681 x^{4}$ |
| Frobenius angles: | $\pm0.0481100458561$, $\pm0.618556620811$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\zeta_{12})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $20$ |
| Cyclic group of points: | yes |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1369$ | $2794129$ | $4685402500$ | $7976346967849$ | $13423232938730809$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $34$ | $1664$ | $67978$ | $2822724$ | $115861154$ | $4749934358$ | $194753391314$ | $7984928807044$ | $327381919270138$ | $13422659102962304$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 20 curves (of which all are hyperelliptic):
- $y^2=32 x^6+16 x^5+5 x^4+18 x^3+39 x^2+5 x+23$
- $y^2=39 x^6+37 x^5+18 x^4+6 x^3+6 x+4$
- $y^2=12 x^6+5 x^5+17 x^4+9 x^3+13 x^2+32 x+31$
- $y^2=8 x^6+13 x^5+4 x^4+31 x^3+26 x^2+26 x+26$
- $y^2=24 x^6+31 x^5+x^3+15 x^2+36 x+14$
- $y^2=29 x^6+40 x^5+37 x^4+25 x^2+24 x+38$
- $y^2=12 x^6+35 x^5+4 x^4+28 x^3+32 x^2+11 x+24$
- $y^2=31 x^6+4 x^5+30 x^4+23 x^3+x^2+36 x+35$
- $y^2=16 x^6+16 x^5+25 x^4+26 x^3+31 x^2+7 x+25$
- $y^2=25 x^6+x^5+26 x^4+29 x^3+6 x^2+11 x+29$
- $y^2=20 x^6+24 x^5+14 x^4+40 x^3+32 x^2+19 x+39$
- $y^2=3 x^6+38 x^5+16 x^4+x^3+18 x^2+12 x+11$
- $y^2=7 x^6+6 x^5+6 x^4+30 x^3+18 x^2+5 x+38$
- $y^2=20 x^6+6 x^5+19 x^4+37 x^3+37 x^2+38 x+30$
- $y^2=14 x^6+13 x^5+24 x^4+13 x^3+39 x^2+4 x+19$
- $y^2=30 x^6+33 x^5+13 x^4+24 x^3+11 x^2+38 x+17$
- $y^2=9 x^6+2 x^5+38 x^3+39 x^2+33 x+19$
- $y^2=20 x^6+15 x^5+14 x^4+16 x^3+10 x^2+36 x+1$
- $y^2=33 x^6+34 x^5+23 x^4+9 x^3+23 x^2+x+32$
- $y^2=11 x^6+16 x^5+9 x^4+9 x^3+39 x^2+33 x+13$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{41^{3}}$.
Endomorphism algebra over $\F_{41}$| The endomorphism algebra of this simple isogeny class is \(\Q(\zeta_{12})\). |
| The base change of $A$ to $\F_{41^{3}}$ is 1.68921.ase 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$ |
Base change
This is a primitive isogeny class.