Invariants
| Base field: | $\F_{2}$ |
| Dimension: | $4$ |
| L-polynomial: | $1 - 3 x + 3 x^{2} - x^{3} - 2 x^{5} + 12 x^{6} - 24 x^{7} + 16 x^{8}$ |
| Frobenius angles: | $\pm0.0463788985841$, $\pm0.192275575577$, $\pm0.470973923744$, $\pm0.819122030567$ |
| Angle rank: | $4$ (numerical) |
| Number field: | 8.0.5198736512.1 |
| Galois group: | $C_2 \wr S_4$ |
| Jacobians: | $1$ |
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: | $3$ |
| Slopes: | $[0, 0, 0, 1/2, 1/2, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $2$ | $124$ | $3482$ | $50096$ | $675422$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $0$ | $2$ | $6$ | $14$ | $20$ | $98$ | $126$ | $222$ | $456$ | $942$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobian of 1 curve (which is not hyperelliptic):
- $x^2+x y+y^2+z t=x^2 y+y^3+x^2 z+z^3+t^3=0$
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 8.0.5198736512.1. |
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 |
|---|---|---|
| 4.2.d_d_b_a | $2$ | 4.4.ad_d_l_abs |