Invariants
| Base field: | $\F_{89}$ |
| Dimension: | $1$ |
| L-polynomial: | $1 + 89 x^{2}$ |
| Frobenius angles: | $\pm0.5$ |
| Angle rank: | $0$ (numerical) |
| Number field: | \(\Q(\sqrt{-89}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $12$ |
| Isomorphism classes: | 12 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $90$ | $8100$ | $704970$ | $62726400$ | $5584059450$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $90$ | $8100$ | $704970$ | $62726400$ | $5584059450$ | $496982700900$ | $44231334895530$ | $3936588680217600$ | $350356403707485210$ | $31181719941134302500$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which 0 are hyperelliptic):
- $y^2=x^3+86 x+86$
- $y^2=x^3+23 x+23$
- $y^2=x^3+75 x+75$
- $y^2=x^3+30 x+30$
- $y^2=x^3+62 x+8$
- $y^2=x^3+3 x+9$
- $y^2=x^3+3$
- $y^2=x^3+19 x+57$
- $y^2=x^3+1$
- $y^2=x^3+12 x+12$
- $y^2=x^3+52 x+67$
- $y^2=x^3+29 x+87$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{89^{2}}$.
Endomorphism algebra over $\F_{89}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-89}) \). |
| The base change of $A$ to $\F_{89^{2}}$ is the simple isogeny class 1.7921.gw and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $89$ and $\infty$. |
Base change
This is a primitive isogeny class.
Twists
This isogeny class has no twists.