Invariants
| Base field: | $\F_{61}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 15 x + 61 x^{2} )( 1 - 13 x + 61 x^{2} )$ |
| $1 - 28 x + 317 x^{2} - 1708 x^{3} + 3721 x^{4}$ | |
| Frobenius angles: | $\pm0.0900194921159$, $\pm0.187058313935$ |
| Angle rank: | $2$ (numerical) |
| Jacobians: | $7$ |
| Cyclic group of points: | yes |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $2303$ | $13299825$ | $51419025728$ | $191735929625625$ | $713383328457705983$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $34$ | $3572$ | $226534$ | $13847908$ | $844644154$ | $51520852262$ | $3142746243874$ | $191707330966468$ | $11694146153878174$ | $713342911718228852$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 7 curves (of which all are hyperelliptic):
- $y^2=20 x^6+x^5+34 x^4+18 x^3+34 x^2+x+20$
- $y^2=52 x^6+56 x^5+55 x^4+14 x^3+55 x^2+56 x+52$
- $y^2=24 x^6+35 x^5+45 x^4+45 x^2+35 x+24$
- $y^2=48 x^6+60 x^5+57 x^4+50 x^3+57 x^2+60 x+48$
- $y^2=6 x^6+49 x^4+22 x^3+49 x^2+6$
- $y^2=44 x^6+45 x^5+25 x^4+45 x^3+25 x^2+45 x+44$
- $y^2=55 x^6+44 x^5+11 x^4+32 x^3+11 x^2+44 x+55$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{61}$.
Endomorphism algebra over $\F_{61}$| The isogeny class factors as 1.61.ap $\times$ 1.61.an 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.