Invariants
| Base field: | $\F_{41}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 + 10 x + 41 x^{2} )( 1 + 12 x + 41 x^{2} )$ |
| $1 + 22 x + 202 x^{2} + 902 x^{3} + 1681 x^{4}$ | |
| Frobenius angles: | $\pm0.785223287477$, $\pm0.886448235704$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $2808$ | $2695680$ | $4751700408$ | $7992152064000$ | $13419705535706808$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $64$ | $1602$ | $68944$ | $2828318$ | $115830704$ | $4750263522$ | $194753577824$ | $7984927070398$ | $327381933681184$ | $13422659313990402$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):
- $y^2=33 x^6+8 x^5+22 x^4+7 x^3+26 x^2+21 x+5$
- $y^2=10 x^6+x^5+40 x^4+11 x^3+40 x^2+x+10$
- $y^2=9 x^6+38 x^5+36 x^4+32 x^3+4 x^2+3 x+40$
- $y^2=9 x^6+6 x^5+19 x^4+18 x^3+19 x^2+6 x+9$
- $y^2=31 x^6+39 x^5+10 x^4+8 x^3+2 x^2+36 x+36$
- $y^2=x^6+9 x^5+30 x^4+13 x^3+30 x^2+9 x+1$
- $y^2=32 x^6+23 x^5+11 x^4+34 x^3+11 x^2+23 x+32$
- $y^2=18 x^6+31 x^5+27 x^4+16 x^3+27 x^2+31 x+18$
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 $\times$ 1.41.m 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.