Invariants
| Base field: | $\F_{61}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 + 12 x + 61 x^{2} )^{2}$ |
| $1 + 24 x + 266 x^{2} + 1464 x^{3} + 3721 x^{4}$ | |
| Frobenius angles: | $\pm0.778857938376$, $\pm0.778857938376$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $9$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2, 37$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $5476$ | $13690000$ | $51308592196$ | $191900067840000$ | $713250098616153316$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $86$ | $3678$ | $226046$ | $13859758$ | $844486406$ | $51520844238$ | $3142743901166$ | $191707271553118$ | $11694146525190326$ | $713342909002702398$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 9 curves (of which all are hyperelliptic):
- $y^2=8 x^6+57 x^5+34 x^4+26 x^3+30 x^2+57 x+60$
- $y^2=12 x^6+45 x^5+56 x^4+30 x^3+31 x^2+32 x+5$
- $y^2=42 x^6+33 x^5+58 x^4+17 x^3+13 x^2+30 x+12$
- $y^2=46 x^6+42 x^5+19 x^4+x^3+2 x^2+27 x+45$
- $y^2=14 x^6+60 x^4+60 x^2+14$
- $y^2=36 x^6+26 x^5+21 x^4+48 x^3+38 x^2+50 x+14$
- $y^2=27 x^6+59 x^5+42 x^4+17 x^3+20 x^2+54 x+28$
- $y^2=53 x^6+13 x^5+2 x^4+46 x^3+31 x^2+58 x+38$
- $y^2=25 x^6+3 x^5+2 x^4+47 x^3+35 x^2+19 x+36$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{61}$.
Endomorphism algebra over $\F_{61}$| The isogeny class factors as 1.61.m 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$ |
Base change
This is a primitive isogeny class.