Invariants
| Base field: | $\F_{7^{2}}$ |
| Dimension: | $1$ |
| L-polynomial: | $1 - 2 x + 49 x^{2}$ |
| Frobenius angles: | $\pm0.454371051657$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $8$ |
| Isomorphism classes: | 8 |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
This isogeny class is simple and geometrically simple, not primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is ordinary.
| $p$-rank: | $1$ |
| Slopes: | $[0, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $48$ | $2496$ | $117936$ | $5760768$ | $282453168$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $48$ | $2496$ | $117936$ | $5760768$ | $282453168$ | $13841440704$ | $678224461872$ | $33232925826048$ | $1628413520361264$ | $79792266374947776$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which 0 are hyperelliptic):
- $y^2=x^3+a^{11} x+a^{12}$
- $y^2=x^3+a^{36} x+a^{36}$
- $y^2=x^3+6 x+6$
- $y^2=x^3+a^{12} x+a^{12}$
- $y^2=x^3+a^{23} x+6$
- $y^2=x^3+a^5 x+a^6$
- $y^2=x^3+a^{17} x+a^{18}$
- $y^2=x^3+1$
where $a$ is a root of the Conway polynomial.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{7^{2}}$.
Endomorphism algebra over $\F_{7^{2}}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}) \). |
Base change
This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{7^{2}}$.
| Subfield | Primitive Model |
| $\F_{7}$ | 1.7.ae |
| $\F_{7}$ | 1.7.e |