Invariants
Base field: | $\F_{2^{6}}$ |
Dimension: | $1$ |
L-polynomial: | $( 1 - 8 x )^{2}$ |
$1 - 16 x + 64 x^{2}$ | |
Frobenius angles: | $0$, $0$ |
Angle rank: | $0$ (numerical) |
Number field: | \(\Q\) |
Galois group: | Trivial |
Jacobians: | $1$ |
This isogeny class is simple and geometrically simple, 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $49$ | $3969$ | $261121$ | $16769025$ | $1073676289$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $49$ | $3969$ | $261121$ | $16769025$ | $1073676289$ | $68718952449$ | $4398042316801$ | $281474943156225$ | $18014398241046529$ | $1152921502459363329$ |
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 $\F_{2^{6}}$.
Endomorphism algebra over $\F_{2^{6}}$The endomorphism algebra of this simple isogeny class is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$. |
Base change
This is a primitive isogeny class.