Invariants
Base field: | $\F_{3^{4}}$ |
Dimension: | $1$ |
L-polynomial: | $1 - 9 x + 81 x^{2}$ |
Frobenius angles: | $\pm0.333333333333$ |
Angle rank: | $0$ (numerical) |
Number field: | \(\Q(\sqrt{-3}) \) |
Galois group: | $C_2$ |
Jacobians: | $1$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $73$ | $6643$ | $532900$ | $43053283$ | $3486725353$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $73$ | $6643$ | $532900$ | $43053283$ | $3486725353$ | $282428473600$ | $22876787671993$ | $1853020231898563$ | $150094636071840100$ | $12157665462543713203$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{3^{12}}$.
Endomorphism algebra over $\F_{3^{4}}$The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}) \). |
The base change of $A$ to $\F_{3^{12}}$ is the simple isogeny class 1.531441.cec and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$. |
Base change
This is a primitive isogeny class.