Invariants
Base field: | $\F_{3^{2}}$ |
Dimension: | $1$ |
L-polynomial: | $1 + 2 x + 9 x^{2}$ |
Frobenius angles: | $\pm0.608173447969$ |
Angle rank: | $1$ (numerical) |
Number field: | \(\Q(\sqrt{-2}) \) |
Galois group: | $C_2$ |
Jacobians: | $3$ |
Isomorphism classes: | 3 |
This isogeny class is simple and geometrically simple, not primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is ordinary.
$p$-rank: | $1$ |
Slopes: | $[0, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $12$ | $96$ | $684$ | $6528$ | $59532$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $12$ | $96$ | $684$ | $6528$ | $59532$ | $530784$ | $4779948$ | $43058688$ | $387423756$ | $3486670176$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{3^{2}}$.
Endomorphism algebra over $\F_{3^{2}}$The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-2}) \). |
Base change
This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{3^{2}}$.
Subfield | Primitive Model |
$\F_{3}$ | 1.3.ac |
$\F_{3}$ | 1.3.c |
Twists
Below is a list of all twists of this isogeny class.
Twist | Extension degree | Common base change |
---|---|---|
1.9.ac | $2$ | 1.81.o |