Invariants
| Base field: | $\F_{71}$ |
| Dimension: | $1$ |
| L-polynomial: | $1 + 71 x^{2}$ |
| Frobenius angles: | $\pm0.5$ |
| Angle rank: | $0$ (numerical) |
| Number field: | \(\Q(\sqrt{-71}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $14$ |
| Isomorphism classes: | 14 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $72$ | $5184$ | $357912$ | $25401600$ | $1804229352$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $72$ | $5184$ | $357912$ | $25401600$ | $1804229352$ | $128100999744$ | $9095120158392$ | $645753480422400$ | $45848500718449032$ | $3255243554618339904$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 14 curves (of which 0 are hyperelliptic):
- $y^2=x^3+22 x+22$
- $y^2=x^3+47 x+45$
- $y^2=x^3+68 x+68$
- $y^2=x^3+13 x+20$
- $y^2=x^3+7$
- $y^2=x^3+63 x+63$
- $y^2=x^3+66 x+36$
- $y^2=x^3+34 x+25$
- $y^2=x^3+x$
- $y^2=x^3+40 x+67$
- $y^2=x^3+14 x+14$
- $y^2=x^3+1$
- $y^2=x^3+7 x$
- $y^2=x^3+24 x+24$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{71^{2}}$.
Endomorphism algebra over $\F_{71}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-71}) \). |
| The base change of $A$ to $\F_{71^{2}}$ is the simple isogeny class 1.5041.fm and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $71$ and $\infty$. |
Base change
This is a primitive isogeny class.
Twists
This isogeny class has no twists.