Invariants
Base field: | $\F_{2^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 4 x^{2} + 16 x^{4}$ |
Frobenius angles: | $\pm0.166666666667$, $\pm0.833333333333$ |
Angle rank: | $0$ (numerical) |
Number field: | \(\Q(\zeta_{12})\) |
Galois group: | $C_2^2$ |
Jacobians: | $1$ |
This isogeny class is simple but not 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: | $[1/2, 1/2, 1/2, 1/2]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $13$ | $169$ | $4225$ | $74529$ | $1047553$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $5$ | $9$ | $65$ | $289$ | $1025$ | $4353$ | $16385$ | $66049$ | $262145$ | $1046529$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:
- $y^2+y=x^5+x^3$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{12}}$.
Endomorphism algebra over $\F_{2^{2}}$The endomorphism algebra of this simple isogeny class is \(\Q(\zeta_{12})\). |
The base change of $A$ to $\F_{2^{12}}$ is 1.4096.ey 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$. |
- Endomorphism algebra over $\F_{2^{4}}$
The base change of $A$ to $\F_{2^{4}}$ is 1.16.ae 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$ - Endomorphism algebra over $\F_{2^{6}}$
The base change of $A$ to $\F_{2^{6}}$ is 1.64.a 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
Base change
This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{2}}$.
Subfield | Primitive Model |
$\F_{2}$ | 2.2.ac_c |
$\F_{2}$ | 2.2.c_c |