Invariants
Base field: | $\F_{5}$ |
Dimension: | $1$ |
L-polynomial: | $1 + 5 x^{2}$ |
Frobenius angles: | $\pm0.5$ |
Angle rank: | $0$ (numerical) |
Number field: | \(\Q(\sqrt{-5}) \) |
Galois group: | $C_2$ |
Jacobians: | $2$ |
Isomorphism classes: | 2 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $6$ | $36$ | $126$ | $576$ | $3126$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $6$ | $36$ | $126$ | $576$ | $3126$ | $15876$ | $78126$ | $389376$ | $1953126$ | $9771876$ |
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 $\F_{5^{2}}$.
Endomorphism algebra over $\F_{5}$The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-5}) \). |
The base change of $A$ to $\F_{5^{2}}$ is the simple isogeny class 1.25.k and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $5$ and $\infty$. |
Base change
This is a primitive isogeny class.
Twists
This isogeny class has no twists.