Invariants
| Base field: | $\F_{7^{2}}$ |
| Dimension: | $1$ |
| L-polynomial: | $1 + 49 x^{2}$ |
| Frobenius angles: | $\pm0.5$ |
| Angle rank: | $0$ (numerical) |
| Number field: | \(\Q(\sqrt{-1}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $2$ |
| Isomorphism classes: | 2 |
| Cyclic group of points: | yes |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $50$ | $2500$ | $117650$ | $5760000$ | $282475250$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $50$ | $2500$ | $117650$ | $5760000$ | $282475250$ | $13841522500$ | $678223072850$ | $33232919040000$ | $1628413597910450$ | $79792266862562500$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which 0 are hyperelliptic):
- $y^2=x^3+a x$
- $y^2=x^3+a^3 x$
where $a$ is a root of the Conway polynomial.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{7^{4}}$.
Endomorphism algebra over $\F_{7^{2}}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-1}) \). |
| The base change of $A$ to $\F_{7^{4}}$ is the simple isogeny class 1.2401.du and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $7$ and $\infty$. |
Base change
This is a primitive isogeny class.
Twists
Below is a list of all twists of this isogeny class.
| Twist | Extension degree | Common base change |
|---|---|---|
| 1.49.ao | $4$ | (not in LMFDB) |
| 1.49.o | $4$ | (not in LMFDB) |