Invariants
| Base field: | $\F_{47}$ |
| Dimension: | $1$ |
| L-polynomial: | $1 + 47 x^{2}$ |
| Frobenius angles: | $\pm0.5$ |
| Angle rank: | $0$ (numerical) |
| Number field: | \(\Q(\sqrt{-47}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $10$ |
| Isomorphism classes: | 10 |
This isogeny class is simple and geometrically simple, primitive, not ordinary, and supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is supersingular.
| $p$-rank: | $0$ |
| Slopes: | $[1/2, 1/2]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $48$ | $2304$ | $103824$ | $4875264$ | $229345008$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $48$ | $2304$ | $103824$ | $4875264$ | $229345008$ | $10779422976$ | $506623120464$ | $23811276902400$ | $1119130473102768$ | $52599132694520064$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which 0 are hyperelliptic):
- $y^2=x^3+14 x+14$
- $y^2=x^3+17 x+38$
- $y^2=x^3+27 x+27$
- $y^2=x^3+6 x+30$
- $y^2=x^3+21 x+11$
- $y^2=x^3+5$
- $y^2=x^3+1$
- $y^2=x^3+5 x$
- $y^2=x^3+4 x+4$
- $y^2=x^3+x$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{47^{2}}$.
Endomorphism algebra over $\F_{47}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-47}) \). |
| The base change of $A$ to $\F_{47^{2}}$ is the simple isogeny class 1.2209.dq and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $47$ and $\infty$. |
Base change
This is a primitive isogeny class.
Twists
This isogeny class has no twists.