Invariants
| Base field: | $\F_{7^{2}}$ |
| Dimension: | $1$ |
| L-polynomial: | $1 - 6 x + 49 x^{2}$ |
| Frobenius angles: | $\pm0.359017035971$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-10}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $6$ |
| Isomorphism classes: | 6 |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $44$ | $2464$ | $118316$ | $5765760$ | $282448364$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $44$ | $2464$ | $118316$ | $5765760$ | $282448364$ | $13841078944$ | $678223140716$ | $33232941181440$ | $1628413658256044$ | $79792266139705504$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which 0 are hyperelliptic):
- $y^2=x^3+a^{12} x+a^{13}$
- $y^2=x^3+a^{43} x+a^{43}$
- $y^2=x^3+6 x+a^{25}$
- $y^2=x^3+a^{37} x+a^{37}$
- $y^2=x^3+a^{19} x+a^{19}$
- $y^2=x^3+a^{13} x+a^{13}$
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 \(\Q(\sqrt{-10}) \). |
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.g | $2$ | (not in LMFDB) |