Invariants
Base field: | $\F_{3^{4}}$ |
Dimension: | $1$ |
L-polynomial: | $( 1 - 9 x )^{2}$ |
$1 - 18 x + 81 x^{2}$ | |
Frobenius angles: | $0$, $0$ |
Angle rank: | $0$ (numerical) |
Number field: | \(\Q\) |
Galois group: | Trivial |
Jacobians: | $1$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $64$ | $6400$ | $529984$ | $43033600$ | $3486666304$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $64$ | $6400$ | $529984$ | $43033600$ | $3486666304$ | $282428473600$ | $22876782889024$ | $1853020102758400$ | $150094634522158144$ | $12157665452083360000$ |
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 $\F_{3^{4}}$.
Endomorphism algebra over $\F_{3^{4}}$The endomorphism algebra of this simple isogeny class is the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$. |
Base change
This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{3^{4}}$.
Subfield | Primitive Model |
$\F_{3}$ | 1.3.a |