Invariants
Base field: | $\F_{3}$ |
Dimension: | $2$ |
L-polynomial: | $1 + 3 x^{2} + 9 x^{4}$ |
Frobenius angles: | $\pm0.333333333333$, $\pm0.666666666667$ |
Angle rank: | $0$ (numerical) |
Number field: | \(\Q(\zeta_{12})\) |
Galois group: | $C_2^2$ |
Jacobians: | $2$ |
Isomorphism classes: | 3 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $13$ | $169$ | $676$ | $8281$ | $59293$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $4$ | $16$ | $28$ | $100$ | $244$ | $622$ | $2188$ | $6724$ | $19684$ | $59536$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=x^6+x^3+2x+1$
- $y^2=2x^6+2x^3+x+2$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{3^{6}}$.
Endomorphism algebra over $\F_{3}$The endomorphism algebra of this simple isogeny class is \(\Q(\zeta_{12})\). |
The base change of $A$ to $\F_{3^{6}}$ is 1.729.acc 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$. |
- Endomorphism algebra over $\F_{3^{2}}$
The base change of $A$ to $\F_{3^{2}}$ is 1.9.d 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$ - Endomorphism algebra over $\F_{3^{3}}$
The base change of $A$ to $\F_{3^{3}}$ is the simple isogeny class 2.27.a_acc and its endomorphism algebra is the quaternion algebra over \(\Q(\sqrt{3}) \) ramified at both real infinite places.
Base change
This is a primitive isogeny class.