Invariants
| Base field: | $\F_{2^{5}}$ |
| Dimension: | $1$ |
| L-polynomial: | $1 - 9 x + 32 x^{2}$ |
| Frobenius angles: | $\pm0.207210850837$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-47}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $5$ |
| Isomorphism classes: | 5 |
| Cyclic group of points: | yes |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $24$ | $1008$ | $32904$ | $1050336$ | $33565944$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $24$ | $1008$ | $32904$ | $1050336$ | $33565944$ | $1073789136$ | $34359795816$ | $1099510630848$ | $35184361278168$ | $1125899841448368$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 5 curves (of which 0 are hyperelliptic):
- $y^2+x y=x^3+a^{25}$
- $y^2+x y=x^3+a^7$
- $y^2+x y=x^3+a^{14}$
- $y^2+x y=x^3+a^{28}$
- $y^2+x y=x^3+a^{19}$
where $a$ is a root of the Conway polynomial.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{5}}$.
Endomorphism algebra over $\F_{2^{5}}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-47}) \). |
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 |
|---|---|---|
| 1.32.j | $2$ | 1.1024.ar |