Invariants
| Base field: | $\F_{61}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 26 x + 285 x^{2} - 1586 x^{3} + 3721 x^{4}$ |
| Frobenius angles: | $\pm0.0471438380502$, $\pm0.263959254932$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-66 +16 \sqrt{6}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $4$ |
| Cyclic group of points: | yes |
This isogeny class is simple and 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $2395$ | $13457505$ | $51496571500$ | $191723273550345$ | $713333008278344875$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $36$ | $3616$ | $226878$ | $13846996$ | $844584576$ | $51519970078$ | $3142737883776$ | $191707276568356$ | $11694145959510198$ | $713342912228782576$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which all are hyperelliptic):
- $y^2=52 x^6+24 x^5+15 x^4+50 x^3+10 x^2+12 x+6$
- $y^2=60 x^6+38 x^5+45 x^4+36 x^3+19 x^2+48 x+40$
- $y^2=25 x^6+4 x^5+4 x^4+50 x^3+31 x^2+20 x+31$
- $y^2=40 x^6+4 x^5+22 x^4+55 x^3+29 x^2+9$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{61}$.
Endomorphism algebra over $\F_{61}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-66 +16 \sqrt{6}})\). |
Base change
This is a primitive isogeny class.
Twists
Below is a list of all twists of this isogeny class.
| Twist | Extension degree | Common base change |
|---|---|---|
| 2.61.ba_kz | $2$ | (not in LMFDB) |