Invariants
Base field: | $\F_{2}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 2 x^{2} )^{2}$ |
$1 - 4 x^{2} + 4 x^{4}$ | |
Frobenius angles: | $0$, $0$, $1$, $1$ |
Angle rank: | $0$ (numerical) |
Number field: | \(\Q(\sqrt{2}) \) |
Galois group: | $C_2$ |
Jacobians: | $0$ |
This isogeny class is simple but not geometrically simple, primitive, not ordinary, and supersingular. It is principally polarizable.
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $1$ | $1$ | $49$ | $81$ | $961$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $3$ | $-3$ | $9$ | $1$ | $33$ | $33$ | $129$ | $193$ | $513$ | $897$ |
Jacobians and polarizations
This isogeny class is principally polarizable, but does not contain a Jacobian.
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 the quaternion algebra over \(\Q(\sqrt{2}) \) ramified at both real infinite places. |
The base change of $A$ to $\F_{2^{2}}$ is 1.4.ae 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$. |
Base change
This is a primitive isogeny class.
Twists
Additional information
This isogeny class appears as a sporadic example in the classification of abelian varieties with one rational point.