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: | $\pm0.214776712523$, $\pm0.214776712523$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $10$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1024$ | $2768896$ | $4781999104$ | $8002109440000$ | $13427514355631104$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $22$ | $1646$ | $69382$ | $2831838$ | $115898102$ | $4750274126$ | $194754254822$ | $7984918073278$ | $327381863616982$ | $13422658895772206$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which all are hyperelliptic):
- $y^2=5 x^6+24 x^4+24 x^2+5$
- $y^2=27 x^6+40 x^5+36 x^4+14 x^3+36 x^2+40 x+27$
- $y^2=38 x^5+22 x^4+13 x^3+12 x^2+13 x$
- $y^2=12 x^6+30 x^4+30 x^2+12$
- $y^2=35 x^6+26 x^5+26 x^4+27 x^3+14 x^2+23 x+7$
- $y^2=29 x^6+27 x^4+27 x^2+29$
- $y^2=3 x^6+23 x^5+32 x^4+11 x^3+32 x^2+23 x+3$
- $y^2=34 x^6+8 x^5+37 x^4+11 x^3+31 x^2+9 x+29$
- $y^2=31 x^6+31 x^5+2 x^4+6 x^3+25 x^2+16 x+36$
- $y^2=13 x^6+13 x^5+30 x^4+2 x^3+30 x^2+13 x+13$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{41}$.
Endomorphism algebra over $\F_{41}$| The isogeny class factors as 1.41.ak 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$ |
Base change
This is a primitive isogeny class.