Invariants
| Base field: | $\F_{61}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 15 x + 61 x^{2} )( 1 - 12 x + 61 x^{2} )$ |
| $1 - 27 x + 302 x^{2} - 1647 x^{3} + 3721 x^{4}$ | |
| Frobenius angles: | $\pm0.0900194921159$, $\pm0.221142061624$ |
| Angle rank: | $2$ (numerical) |
| Jacobians: | $5$ |
| Isomorphism classes: | 13 |
| Cyclic group of points: | no |
| Non-cyclic primes: | $5$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $2350$ | $13390300$ | $51483762400$ | $191759808240000$ | $713381657019583750$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $35$ | $3597$ | $226820$ | $13849633$ | $844642175$ | $51520666362$ | $3142743712955$ | $191707309936993$ | $11694146055603140$ | $713342911939629477$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 5 curves (of which all are hyperelliptic):
- $y^2=16 x^6+22 x^5+3 x^4+43 x^3+21 x^2+50 x+43$
- $y^2=29 x^6+21 x^5+35 x^4+47 x^3+12 x^2+12 x+37$
- $y^2=38 x^6+43 x^5+3 x^4+2 x^3+8 x^2+36 x+56$
- $y^2=43 x^6+34 x^5+21 x^4+49 x^2+34 x+24$
- $y^2=21 x^6+37 x^5+21 x^4+57 x^3+9 x^2+41 x+41$
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.am 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.