Invariants
| Base field: | $\F_{41}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - x + 41 x^{2} )^{2}$ |
| $1 - 2 x + 83 x^{2} - 82 x^{3} + 1681 x^{4}$ | |
| Frobenius angles: | $\pm0.475118844265$, $\pm0.475118844265$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $7$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $41$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1681$ | $3108169$ | $4767073936$ | $7966861888969$ | $13420759335696001$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $40$ | $1844$ | $69166$ | $2819364$ | $115839800$ | $4750350158$ | $194755192280$ | $7984916064964$ | $327381887575486$ | $13422659639064404$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 7 curves (of which all are hyperelliptic):
- $y^2=39 x^6+16 x^5+15 x^4+x^3+24 x^2+18 x+21$
- $y^2=9 x^6+16 x^5+39 x^4+39 x^3+5 x^2+32 x+6$
- $y^2=28 x^6+30 x^5+4 x^4+16 x^2+32 x+40$
- $y^2=30 x^6+20 x^5+2 x^4+27 x^3+32 x^2+23 x+29$
- $y^2=39 x^6+10 x^5+17 x^4+26 x^3+31 x^2+28 x+3$
- $y^2=28 x^6+6 x^5+7 x^4+4 x^3+22 x^2+13 x+37$
- $y^2=39 x^6+29 x^5+19 x^4+25 x^3+30 x^2+10 x+5$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{41}$.
Endomorphism algebra over $\F_{41}$| The isogeny class factors as 1.41.ab 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-163}) \)$)$ |
Base change
This is a primitive isogeny class.