Invariants
| Base field: | $\F_{2}$ |
| Dimension: | $6$ |
| L-polynomial: | $1 + 3 x^{2} + x^{3} + 6 x^{4} + 6 x^{5} + 8 x^{6} + 12 x^{7} + 24 x^{8} + 8 x^{9} + 48 x^{10} + 64 x^{12}$ |
| Frobenius angles: | $\pm0.182860320188$, $\pm0.308818668718$, $\pm0.413609783026$, $\pm0.596254109314$, $\pm0.645859071324$, $\pm0.882418177107$ |
| Angle rank: | $6$ (numerical) |
| Number field: | 12.0.188838465943828032.1 |
| Galois group: | $D_4\wr C_3$ |
| Jacobians: | $1$ |
| Cyclic group of points: | yes |
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: | $[0, 0, 0, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $181$ | $22987$ | $339013$ | $29814139$ | $1665734131$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $3$ | $11$ | $12$ | $23$ | $48$ | $56$ | $108$ | $359$ | $435$ | $956$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobian of 1 curve (which is not hyperelliptic):
- $x^{8} + x^{7} y + x^{6} y^{2} + x^{4} y^{4} + x^{2} y^{6} + x^{7} + x^{6} y + x^{4} y^{3} + x y^{6} + x^{6} + x^{4} y^{2} + x^{2} y^{4} + x^{3} y^{2} + x^{2} y^{3} + x y^{4} + x^{4} + x^{3} y + x y^{3} + x^{3} + x^{2} y + x y^{2} + x^{2} + x y + y^{2} + x + y + 1=0$
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 12.0.188838465943828032.1. |
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 |
|---|---|---|
| 6.2.a_d_ab_g_ag_i | $2$ | (not in LMFDB) |