Invariants
| Base field: | $\F_{7}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 2 x^{2} + 49 x^{4}$ |
| Frobenius angles: | $\pm0.272814474171$, $\pm0.727185525829$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\zeta_{12})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $10$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $52$ | $2704$ | $117364$ | $6230016$ | $282497332$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $8$ | $54$ | $344$ | $2590$ | $16808$ | $117078$ | $823544$ | $5756734$ | $40353608$ | $282519414$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which all are hyperelliptic):
- $y^2=5 x^6+5 x^4+5 x^3+x^2+5 x$
- $y^2=x^6+x^4+x^3+3 x^2+x$
- $y^2=x^6+4 x^5+3 x^4+4 x^3+3 x^2+x$
- $y^2=5 x^6+6 x^5+x^4+6 x^3+3 x^2+5 x+2$
- $y^2=6 x^6+3 x^5+3 x^3+5 x+1$
- $y^2=x^6+6 x^5+2 x^4+4 x^3+5 x^2+6 x+6$
- $y^2=x^6+4 x^5+5 x^4+5 x^3+3 x^2+x+6$
- $y^2=5 x^6+4 x^5+3 x^4+x^3+x^2+2 x+2$
- $y^2=5 x^6+x^5+x^4+4 x^3+5 x$
- $y^2=x^6+3 x^5+3 x^4+5 x^3+x$
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 \(\Q(\zeta_{12})\). |
| The base change of $A$ to $\F_{7^{2}}$ is 1.49.c 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$ |
Base change
This is a primitive isogeny class.