Invariants
| Base field: | $\F_{61}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 8 x + 3 x^{2} + 488 x^{3} + 3721 x^{4}$ |
| Frobenius angles: | $\pm0.337816561762$, $\pm0.995516771572$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}, \sqrt{-5})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $10$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $4221$ | $13629609$ | $51953908356$ | $191650852437849$ | $713364404752580301$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $70$ | $3664$ | $228886$ | $13841764$ | $844621750$ | $51519469678$ | $3142744902430$ | $191707301935684$ | $11694146521921486$ | $713342910621290704$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which all are hyperelliptic):
- $y^2=56 x^6+58 x^5+52 x^4+11 x^3+3 x^2+56 x+29$
- $y^2=47 x^6+43 x^5+55 x^4+8 x^3+42 x^2+19 x+36$
- $y^2=x^6+x^3+16$
- $y^2=x^6+2 x^3+19$
- $y^2=11 x^6+3 x^5+23 x^4+15 x^3+41 x^2+4 x+58$
- $y^2=x^6+x^3+39$
- $y^2=21 x^6+7 x^5+3 x^4+12 x^3+35 x^2+17 x+54$
- $y^2=x^6+x^3+36$
- $y^2=x^6+x^3+25$
- $y^2=x^6+x^3+56$
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.bkq 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-5}) \)$)$ |
Base change
This is a primitive isogeny class.