Invariants
| Base field: | $\F_{61}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 8 x + 3 x^{2} - 488 x^{3} + 3721 x^{4}$ |
| Frobenius angles: | $\pm0.00448322842836$, $\pm0.662183438238$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}, \sqrt{-5})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $10$ |
| Isomorphism classes: | 14 |
| Cyclic group of points: | yes |
This isogeny class is simple but not geometrically 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $3229$ | $13629609$ | $51089560900$ | $191650852437849$ | $713321418179193709$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $54$ | $3664$ | $225078$ | $13841764$ | $844570854$ | $51519469678$ | $3142740769614$ | $191707301935684$ | $11694145663746798$ | $713342910621290704$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which all are hyperelliptic):
- $y^2=28 x^6+29 x^5+26 x^4+36 x^3+32 x^2+28 x+45$
- $y^2=33 x^6+25 x^5+49 x^4+16 x^3+23 x^2+38 x+11$
- $y^2=2 x^6+2 x^3+32$
- $y^2=2 x^6+4 x^3+38$
- $y^2=36 x^6+32 x^5+42 x^4+38 x^3+51 x^2+2 x+29$
- $y^2=2 x^6+2 x^3+17$
- $y^2=42 x^6+14 x^5+6 x^4+24 x^3+9 x^2+34 x+47$
- $y^2=2 x^6+2 x^3+11$
- $y^2=2 x^6+2 x^3+50$
- $y^2=2 x^6+2 x^3+51$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{61^{3}}$.
Endomorphism algebra over $\F_{61}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-5})\). |
| The base change of $A$ to $\F_{61^{3}}$ is 1.226981.abkq 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-5}) \)$)$ |
Base change
This is a primitive isogeny class.