Invariants
| Base field: | $\F_{2}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 - 2 x + 3 x^{2} - 6 x^{3} + 6 x^{4} - 8 x^{5} + 8 x^{6}$ |
| Frobenius angles: | $\pm0.0325998547649$, $\pm0.440077654430$, $\pm0.657477799665$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 6.0.399424.1 |
| Galois group: | $D_{6}$ |
| Jacobians: | $1$ |
| Isomorphism classes: | 1 |
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: | $2$ |
| Slopes: | $[0, 0, 1/2, 1/2, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $2$ | $68$ | $146$ | $2176$ | $22082$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $1$ | $7$ | $1$ | $7$ | $21$ | $43$ | $141$ | $255$ | $397$ | $987$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:
- $y^2+(x^2+x+1)y=x^7+x^6+x^5+x^4+x^3+x+1$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2}$.
Endomorphism algebra over $\F_{2}$| The endomorphism algebra of this simple isogeny class is 6.0.399424.1. |
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 |
|---|---|---|
| 3.2.c_d_g | $2$ | 3.4.c_ad_aq |
| 3.2.a_ab_ac | $4$ | 3.16.ak_bx_ahc |
| 3.2.a_ab_c | $4$ | 3.16.ak_bx_ahc |