Invariants
| Base field: | $\F_{7^{2}}$ |
| Dimension: | $1$ |
| L-polynomial: | $1 - 5 x + 49 x^{2}$ |
| Frobenius angles: | $\pm0.383750930958$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-19}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $5$ |
| Isomorphism classes: | 5 |
| Cyclic group of points: | no |
| Non-cyclic primes: | $3$ |
This isogeny class is simple and geometrically simple, 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $45$ | $2475$ | $118260$ | $5764275$ | $282442725$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $45$ | $2475$ | $118260$ | $5764275$ | $282442725$ | $13841150400$ | $678223982565$ | $33232941821475$ | $1628413609593780$ | $79792265804686875$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 5 curves (of which 0 are hyperelliptic):
- $y^2=x^3+a^2 x+a^3$
- $y^2=x^3+4 x+a^{33}$
- $y^2=x^3+a^{39} x+a^{39}$
- $y^2=x^3+a^{33} x+a^{33}$
- $y^2=x^3+a^{20} x+a^{21}$
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 \(\Q(\sqrt{-19}) \). |
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.f | $2$ | (not in LMFDB) |