Invariants
| Base field: | $\F_{41}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 12 x + 41 x^{2} )( 1 - 10 x + 41 x^{2} )$ |
| $1 - 22 x + 202 x^{2} - 902 x^{3} + 1681 x^{4}$ | |
| Frobenius angles: | $\pm0.113551764296$, $\pm0.214776712523$ |
| Angle rank: | $2$ (numerical) |
| Jacobians: | $8$ |
| Isomorphism classes: | 30 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $960$ | $2695680$ | $4748667840$ | $7992152064000$ | $13425613738584000$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $20$ | $1602$ | $68900$ | $2828318$ | $115881700$ | $4750263522$ | $194754969940$ | $7984927070398$ | $327381935106740$ | $13422659313990402$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=34x^6+7x^5+9x^4+x^3+33x^2+3x+30$
- $y^2=19x^6+6x^5+35x^4+25x^3+35x^2+6x+19$
- $y^2=13x^6+23x^5+11x^4+28x^3+24x^2+18x+35$
- $y^2=13x^6+36x^5+32x^4+26x^3+32x^2+36x+13$
- $y^2=12x^6+27x^5+29x^4+15x^3+14x^2+6x+6$
- $y^2=7x^6+22x^5+5x^4+9x^3+5x^2+22x+7$
- $y^2=28x^6+15x^5+25x^4+40x^3+25x^2+15x+28$
- $y^2=26x^6+22x^5+39x^4+14x^3+39x^2+22x+26$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{41}$.
Endomorphism algebra over $\F_{41}$| The isogeny class factors as 1.41.am $\times$ 1.41.ak and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
Base change
This is a primitive isogeny class.