Invariants
| Base field: | $\F_{41}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 8 x + 41 x^{2} )^{2}$ |
| $1 - 16 x + 146 x^{2} - 656 x^{3} + 1681 x^{4}$ | |
| Frobenius angles: | $\pm0.285223287477$, $\pm0.285223287477$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $9$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2, 17$ |
This isogeny class is not 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1156$ | $2890000$ | $4815527236$ | $8002109440000$ | $13423807006211716$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $26$ | $1718$ | $69866$ | $2831838$ | $115866106$ | $4749934358$ | $194752508746$ | $7984918073278$ | $327381949517786$ | $13422659724532598$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 9 curves (of which all are hyperelliptic):
- $y^2=26 x^6+19 x^5+18 x^4+26 x^3+33 x^2+24 x+30$
- $y^2=29 x^6+12 x^5+15 x^4+17 x^3+16 x^2+19 x+27$
- $y^2=4 x^6+37 x^5+3 x^4+7 x^3+28 x^2+16 x+1$
- $y^2=2 x^6+2 x^5+28 x^4+28 x^3+19 x^2+5 x+40$
- $y^2=31 x^6+x^5+35 x^4+x^3+11 x^2+25 x+21$
- $y^2=12 x^6+35 x^5+21 x^4+22 x^3+28 x^2+24 x+27$
- $y^2=38 x^6+22 x^5+22 x^4+33 x^3+25 x^2+38 x+37$
- $y^2=26 x^6+x^4+x^2+26$
- $y^2=5 x^6+29 x^5+4 x^4+14 x^3+34 x^2+34 x+6$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{41}$.
Endomorphism algebra over $\F_{41}$| The isogeny class factors as 1.41.ai 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$ |
Base change
This is a primitive isogeny class.