Invariants
| Base field: | $\F_{53}$ |
| Dimension: | $1$ |
| L-polynomial: | $1 + 53 x^{2}$ |
| Frobenius angles: | $\pm0.5$ |
| Angle rank: | $0$ (numerical) |
| Number field: | \(\Q(\sqrt{-53}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $6$ |
| Isomorphism classes: | 6 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $54$ | $2916$ | $148878$ | $7884864$ | $418195494$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $54$ | $2916$ | $148878$ | $7884864$ | $418195494$ | $22164658884$ | $1174711139838$ | $62259674630400$ | $3299763591802134$ | $174887471201904036$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which 0 are hyperelliptic):
- $y^2=x^3+2$
- $y^2=x^3+21 x+21$
- $y^2=x^3+31 x+9$
- $y^2=x^3+40 x+27$
- $y^2=x^3+1$
- $y^2=x^3+10 x+10$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{53^{2}}$.
Endomorphism algebra over $\F_{53}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-53}) \). |
| The base change of $A$ to $\F_{53^{2}}$ is the simple isogeny class 1.2809.ec and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $53$ and $\infty$. |
Base change
This is a primitive isogeny class.
Twists
This isogeny class has no twists.