Invariants
Base field: | $\F_{2}$ |
Dimension: | $2$ |
L-polynomial: | $1 + x - x^{2} + 2 x^{3} + 4 x^{4}$ |
Frobenius angles: | $\pm0.281693394748$, $\pm0.948360061415$ |
Angle rank: | $1$ (numerical) |
Number field: | \(\Q(\sqrt{-3}, \sqrt{-7})\) |
Galois group: | $C_2^2$ |
Jacobians: | $0$ |
Isomorphism classes: | 1 |
This isogeny class is simple but not geometrically simple, primitive, ordinary, and not supersingular.
Newton polygon
This isogeny class is ordinary.
$p$-rank: | $2$ |
Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $7$ | $7$ | $196$ | $259$ | $1477$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $4$ | $2$ | $19$ | $18$ | $44$ | $47$ | $116$ | $226$ | $523$ | $1082$ |
Jacobians and polarizations
This isogeny class is not principally polarizable, and therefore does not contain a Jacobian.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{3}}$.
Endomorphism algebra over $\F_{2}$The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-7})\). |
The base change of $A$ to $\F_{2^{3}}$ is 1.8.f 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$ |
Base change
This is a primitive isogeny class.