Invariants
| Base field: | $\F_{7}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 49 x^{4}$ |
| Frobenius angles: | $\pm0.250000000000$, $\pm0.750000000000$ |
| Angle rank: | $0$ (numerical) |
| Number field: | \(\Q(i, \sqrt{14})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $9$ |
This isogeny class is simple but not 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: | $[1/2, 1/2, 1/2, 1/2]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $50$ | $2500$ | $117650$ | $6250000$ | $282475250$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $8$ | $50$ | $344$ | $2598$ | $16808$ | $117650$ | $823544$ | $5755198$ | $40353608$ | $282475250$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 9 curves (of which all are hyperelliptic):
- $y^2=x^5+6$
- $y^2=3 x^5+4$
- $y^2=5 x^6+5 x^5+4 x^3+6 x^2+3 x+1$
- $y^2=x^6+x^5+5 x^3+4 x^2+2 x+3$
- $y^2=3 x^6+5 x^5+5 x^4+4 x^3+2 x^2+3 x+6$
- $y^2=2 x^6+x^5+x^4+5 x^3+6 x^2+2 x+4$
- $y^2=5 x^6+5 x^5+6 x^3+x+2$
- $y^2=x^6+x^5+4 x^3+3 x+6$
- $y^2=x^6+5 x^5+2 x^4+x^3+x^2+4 x+3$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{7^{4}}$.
Endomorphism algebra over $\F_{7}$| The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{14})\). |
| The base change of $A$ to $\F_{7^{4}}$ is 1.2401.du 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $7$ and $\infty$. |
- Endomorphism algebra over $\F_{7^{2}}$
The base change of $A$ to $\F_{7^{2}}$ is 1.49.a 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
Base change
This is a primitive isogeny class.