Invariants
| Base field: | $\F_{5}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 5 x^{2} + 25 x^{4}$ |
| Frobenius angles: | $\pm0.166666666667$, $\pm0.833333333333$ |
| Angle rank: | $0$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}, \sqrt{-5})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $1$ |
| Isomorphism classes: | 6 |
This isogeny class is simple but not 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: | $[1/2, 1/2, 1/2, 1/2]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $21$ | $441$ | $15876$ | $423801$ | $9762501$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $6$ | $16$ | $126$ | $676$ | $3126$ | $16126$ | $78126$ | $391876$ | $1953126$ | $9759376$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobian of 1 curve (which is hyperelliptic):
- $y^2=x^6+x^5+2$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{5^{6}}$.
Endomorphism algebra over $\F_{5}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-5})\). |
| The base change of $A$ to $\F_{5^{6}}$ is 1.15625.jq 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $5$ and $\infty$. |
- Endomorphism algebra over $\F_{5^{2}}$
The base change of $A$ to $\F_{5^{2}}$ is 1.25.af 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$ - Endomorphism algebra over $\F_{5^{3}}$
The base change of $A$ to $\F_{5^{3}}$ is 1.125.a 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-5}) \)$)$
Base change
This is a primitive isogeny class.