Invariants
| Base field: | $\F_{5^{3}}$ |
| Dimension: | $1$ |
| L-polynomial: | $1 + 125 x^{2}$ |
| Frobenius angles: | $\pm0.5$ |
| Angle rank: | $0$ (numerical) |
| Number field: | \(\Q(\sqrt{-5}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $2$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $126$ | $15876$ | $1953126$ | $244109376$ | $30517578126$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $126$ | $15876$ | $1953126$ | $244109376$ | $30517578126$ | $3814701171876$ | $476837158203126$ | $59604644287109376$ | $7450580596923828126$ | $931322574676513671876$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{5^{6}}$.
Endomorphism algebra over $\F_{5^{3}}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-5}) \). |
| The base change of $A$ to $\F_{5^{6}}$ is the simple isogeny class 1.15625.jq and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $5$ and $\infty$. |
Base change
This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{5^{3}}$.
| Subfield | Primitive Model |
| $\F_{5}$ | 1.5.a |
Twists
This isogeny class has no twists.