Invariants
| Base field: | $\F_{7}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 5 x + 7 x^{2} )( 1 + 5 x + 7 x^{2} )$ |
| $1 - 11 x^{2} + 49 x^{4}$ | |
| Frobenius angles: | $\pm0.106147807505$, $\pm0.893852192495$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $1$ |
| Cyclic group of points: | yes |
This isogeny class is not 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $39$ | $1521$ | $117936$ | $5659641$ | $282508239$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $8$ | $28$ | $344$ | $2356$ | $16808$ | $118222$ | $823544$ | $5773348$ | $40353608$ | $282541228$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobian of 1 curve (which is hyperelliptic):
- $y^2=x^6+5 x^3+6$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{7^{2}}$.
Endomorphism algebra over $\F_{7}$| The isogeny class factors as 1.7.af $\times$ 1.7.f and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
| The base change of $A$ to $\F_{7^{2}}$ is 1.49.al 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$ |
Base change
This is a primitive isogeny class.