Invariants
Base field: | $\F_{7}$ |
Dimension: | $1$ |
L-polynomial: | $1 + 7 x^{2}$ |
Frobenius angles: | $\pm0.5$ |
Angle rank: | $0$ (numerical) |
Number field: | \(\Q(\sqrt{-7}) \) |
Galois group: | $C_2$ |
Jacobians: | $2$ |
Isomorphism classes: | 2 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $8$ | $64$ | $344$ | $2304$ | $16808$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $8$ | $64$ | $344$ | $2304$ | $16808$ | $118336$ | $823544$ | $5760000$ | $40353608$ | $282508864$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{7^{2}}$.
Endomorphism algebra over $\F_{7}$The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-7}) \). |
The base change of $A$ to $\F_{7^{2}}$ is the simple isogeny class 1.49.o and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $7$ and $\infty$. |
Base change
This is a primitive isogeny class.
Twists
This isogeny class has no twists.