Invariants
| Base field: | $\F_{2}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 - 2 x^{3} + 8 x^{6}$ |
| Frobenius angles: | $\pm0.128324423973$, $\pm0.538342242694$, $\pm0.794991090640$ |
| Angle rank: | $1$ (numerical) |
| Number field: | 6.0.4000752.1 |
| Galois group: | $D_{6}$ |
| Jacobians: | $1$ |
| Isomorphism classes: | 2 |
This isogeny class is simple and geometrically simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
| $p$-rank: | $0$ |
| Slopes: | $[1/3, 1/3, 1/3, 2/3, 2/3, 2/3]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $7$ | $77$ | $343$ | $4081$ | $32417$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $3$ | $5$ | $3$ | $17$ | $33$ | $101$ | $129$ | $257$ | $633$ | $1025$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:
- $y^2+y=x^7$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{3}}$.
Endomorphism algebra over $\F_{2}$| The endomorphism algebra of this simple isogeny class is 6.0.4000752.1. |
| The base change of $A$ to $\F_{2^{3}}$ is the simple isogeny class 3.8.ag_bk_aea and its endomorphism algebra is the division algebra of dimension 9 over \(\Q(\sqrt{-7}) \) with the following ramification data at primes above $2$, and unramified at all archimedean places: | ||||||
|
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 |
|---|---|---|
| 3.2.a_a_c | $2$ | 3.4.a_a_m |
| 3.2.a_a_c | $6$ | (not in LMFDB) |