Invariants
| Base field: | $\F_{73}$ |
| Dimension: | $1$ |
| L-polynomial: | $1 + 73 x^{2}$ |
| Frobenius angles: | $\pm0.5$ |
| Angle rank: | $0$ (numerical) |
| Number field: | \(\Q(\sqrt{-73}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $4$ |
| Isomorphism classes: | 4 |
| Cyclic group of points: | yes |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $74$ | $5476$ | $389018$ | $28387584$ | $2073071594$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $74$ | $5476$ | $389018$ | $28387584$ | $2073071594$ | $151335004324$ | $11047398519098$ | $806460035097600$ | $58871586708267914$ | $4297625833849700836$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which 0 are hyperelliptic):
- $y^2=x^3+22 x+37$
- $y^2=x^3+37 x+39$
- $y^2=x^3+19 x+19$
- $y^2=x^3+33 x+33$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{73^{2}}$.
Endomorphism algebra over $\F_{73}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-73}) \). |
| The base change of $A$ to $\F_{73^{2}}$ is the simple isogeny class 1.5329.fq and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $73$ and $\infty$. |
Base change
This is a primitive isogeny class.
Twists
This isogeny class has no twists.