Invariants
| Base field: | $\F_{2^{3}}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 - 10 x + 50 x^{2} - 167 x^{3} + 400 x^{4} - 640 x^{5} + 512 x^{6}$ |
| Frobenius angles: | $\pm0.0706215684044$, $\pm0.235483793327$, $\pm0.482907879127$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.386310791.1 |
| Galois group: | $S_4\times C_2$ |
| Isomorphism classes: | 15 |
This isogeny class is simple and geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable.
Newton polygon
This isogeny class is ordinary.
| $p$-rank: | $3$ |
| Slopes: | $[0, 0, 0, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $146$ | $259880$ | $134205536$ | $67363494800$ | $35239205936266$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $-1$ | $65$ | $512$ | $4017$ | $32819$ | $263150$ | $2096863$ | $16763777$ | $134194976$ | $1073808825$ |
Jacobians and polarizations
This isogeny class is principally polarizable, but does not contain a Jacobian.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{3}}$.
Endomorphism algebra over $\F_{2^{3}}$| The endomorphism algebra of this simple isogeny class is 6.0.386310791.1. |
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 |
|---|---|---|
| 3.8.k_by_gl | $2$ | (not in LMFDB) |