Invariants
| Base field: | $\F_{449}$ |
| Dimension: | $1$ |
| L-polynomial: | $1 + 449 x^{2}$ |
| Frobenius angles: | $\pm0.5$ |
| Angle rank: | $0$ (numerical) |
| Number field: | \(\Q(\sqrt{-449}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $20$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $450$ | $202500$ | $90518850$ | $40642560000$ | $18248690477250$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $450$ | $202500$ | $90518850$ | $40642560000$ | $18248690477250$ | $8193662205322500$ | $3678954248903875650$ | $1651850457676554240000$ | $741680855533270234714050$ | $333014704134474832767562500$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 20 curves (of which all are hyperelliptic), and hence is principally polarizable:
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{449^{2}}$.
Endomorphism algebra over $\F_{449}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-449}) \). |
| The base change of $A$ to $\F_{449^{2}}$ is the simple isogeny class 1.201601.bio and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $449$ and $\infty$. |
Base change
This is a primitive isogeny class.
Twists
This isogeny class has no twists.