Invariants
| Base field: | $\F_{2^{2}}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 - 4 x + 13 x^{2} - 28 x^{3} + 52 x^{4} - 64 x^{5} + 64 x^{6}$ |
| Frobenius angles: | $\pm0.185044400669$, $\pm0.380155308860$, $\pm0.565199709530$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 6.0.399424.1 |
| Galois group: | $D_{6}$ |
| 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: | $2$ |
| Slopes: | $[0, 0, 1/2, 1/2, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $34$ | $7684$ | $300322$ | $16720384$ | $1131706354$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $1$ | $27$ | $73$ | $255$ | $1081$ | $4179$ | $16409$ | $66111$ | $262873$ | $1044147$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobian of 1 curve (which is hyperelliptic):
- $y^2+((a+1) x^4+a x^3+a x^2) y=x^8+(a+1) x^3+(a+1) x$
where $a$ is a root of the Conway polynomial.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{2}}$.
Endomorphism algebra over $\F_{2^{2}}$| 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 |
|---|---|---|
| 3.4.e_n_bc | $2$ | 3.16.k_bx_hc |
| 3.4.ac_ad_q | $4$ | (not in LMFDB) |
| 3.4.c_ad_aq | $4$ | (not in LMFDB) |