Invariants
Base field: | $\F_{2}$ |
Dimension: | $2$ |
L-polynomial: | $1 - x^{2} + 4 x^{4}$ |
Frobenius angles: | $\pm0.209784688372$, $\pm0.790215311628$ |
Angle rank: | $1$ (numerical) |
Number field: | \(\Q(\sqrt{-3}, \sqrt{5})\) |
Galois group: | $C_2^2$ |
Jacobians: | $1$ |
Isomorphism classes: | 4 |
This isogeny class is simple but not geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is ordinary.
$p$-rank: | $2$ |
Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $4$ | $16$ | $76$ | $576$ | $964$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $3$ | $3$ | $9$ | $31$ | $33$ | $87$ | $129$ | $223$ | $513$ | $903$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:
- $y^2+(x^2+x)y=x^5+x^3+x^2+x$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{2}}$.
Endomorphism algebra over $\F_{2}$The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{5})\). |
The base change of $A$ to $\F_{2^{2}}$ is 1.4.ab 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$ |
Base change
This is a primitive isogeny class.