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: | $10$ |
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 contains the Jacobians of 10 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=5x^6+24x^4+24x^2+5$
- $y^2=27x^6+40x^5+36x^4+14x^3+36x^2+40x+27$
- $y^2=38x^5+22x^4+13x^3+12x^2+13x$
- $y^2=12x^6+30x^4+30x^2+12$
- $y^2=35x^6+26x^5+26x^4+27x^3+14x^2+23x+7$
- $y^2=29x^6+27x^4+27x^2+29$
- $y^2=3x^6+23x^5+32x^4+11x^3+32x^2+23x+3$
- $y^2=34x^6+8x^5+37x^4+11x^3+31x^2+9x+29$
- $y^2=31x^6+31x^5+2x^4+6x^3+25x^2+16x+36$
- $y^2=13x^6+13x^5+30x^4+2x^3+30x^2+13x+13$
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.