Invariants
| Base field: | $\F_{41}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 + 10 x + 41 x^{2} )^{2}$ |
| $1 + 20 x + 182 x^{2} + 820 x^{3} + 1681 x^{4}$ | |
| Frobenius angles: | $\pm0.785223287477$, $\pm0.785223287477$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $10$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $2704$ | $2768896$ | $4718590864$ | $8002109440000$ | $13417805605903504$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $62$ | $1646$ | $68462$ | $2831838$ | $115814302$ | $4750274126$ | $194754292942$ | $7984918073278$ | $327382005170942$ | $13422658895772206$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which all are hyperelliptic):
- $y^2=30 x^6+21 x^4+21 x^2+30$
- $y^2=25 x^6+34 x^5+6 x^4+16 x^3+6 x^2+34 x+25$
- $y^2=23 x^5+9 x^4+37 x^3+31 x^2+37 x$
- $y^2=31 x^6+16 x^4+16 x^2+31$
- $y^2=5 x^6+33 x^5+33 x^4+39 x^3+2 x^2+15 x+1$
- $y^2=10 x^6+39 x^4+39 x^2+10$
- $y^2=18 x^6+15 x^5+28 x^4+25 x^3+28 x^2+15 x+18$
- $y^2=33 x^6+15 x^5+13 x^4+36 x^3+12 x^2+22 x+39$
- $y^2=12 x^6+12 x^5+14 x^4+x^3+11 x^2+30 x+6$
- $y^2=37 x^6+37 x^5+16 x^4+12 x^3+16 x^2+37 x+37$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{41}$.
Endomorphism algebra over $\F_{41}$| The isogeny class factors as 1.41.k 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$ |
Base change
This is a primitive isogeny class.