Invariants
| Base field: | $\F_{2^{9}}$ |
| Dimension: | $1$ |
| L-polynomial: | $1 + 512 x^{2}$ |
| Frobenius angles: | $\pm0.5$ |
| Angle rank: | $0$ (numerical) |
| Number field: | \(\Q(\sqrt{-2}) \) |
| Galois group: | $C_2$ |
This isogeny class is simple and geometrically simple, not 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $513$ | $263169$ | $134217729$ | $68718952449$ | $35184372088833$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $513$ | $263169$ | $134217729$ | $68718952449$ | $35184372088833$ | $18014398777917441$ | $9223372036854775809$ | $4722366482732206260225$ | $2417851639229258349412353$ | $1237940039285450643643301889$ |
Jacobians and polarizations
This isogeny class contains a Jacobian, and hence is principally polarizable.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{18}}$.
Endomorphism algebra over $\F_{2^{9}}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-2}) \). |
| The base change of $A$ to $\F_{2^{18}}$ is the simple isogeny class 1.262144.bnk and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$. |
Base change
This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{9}}$.
| Subfield | Primitive Model |
| $\F_{2}$ | 1.2.a |
Twists
Below is a list of all twists of this isogeny class.
| Twist | Extension degree | Common base change |
|---|---|---|
| 1.512.abg | $8$ | (not in LMFDB) |
| 1.512.bg | $8$ | (not in LMFDB) |