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: | $\pm0.714776712523$, $\pm0.714776712523$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $9$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2, 5$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $2500$ | $2890000$ | $4685402500$ | $8002109440000$ | $13421512126562500$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $58$ | $1718$ | $67978$ | $2831838$ | $115846298$ | $4749934358$ | $194756039018$ | $7984918073278$ | $327381919270138$ | $13422659724532598$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 9 curves (of which all are hyperelliptic):
- $y^2=18 x^6+10 x^5+3 x^4+18 x^3+26 x^2+4 x+5$
- $y^2=10 x^6+31 x^5+8 x^4+20 x^3+14 x^2+32 x+39$
- $y^2=24 x^6+17 x^5+18 x^4+x^3+4 x^2+14 x+6$
- $y^2=14 x^6+14 x^5+32 x^4+32 x^3+10 x^2+35 x+34$
- $y^2=12 x^6+7 x^5+40 x^4+7 x^3+36 x^2+11 x+24$
- $y^2=31 x^6+5 x^5+3 x^4+9 x^3+4 x^2+21 x+39$
- $y^2=20 x^6+31 x^5+31 x^4+26 x^3+11 x^2+20 x+13$
- $y^2=33 x^6+6 x^4+6 x^2+33$
- $y^2=30 x^6+10 x^5+24 x^4+2 x^3+40 x^2+40 x+36$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{41}$.
Endomorphism algebra over $\F_{41}$| The isogeny class factors as 1.41.i 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$ |
Base change
This is a primitive isogeny class.