Invariants
| Base field: | $\F_{2^{2}}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - x + 7 x^{2} - 4 x^{3} + 16 x^{4}$ |
| Frobenius angles: | $\pm0.367442607140$, $\pm0.549379399016$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.5225.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $2$ |
| Isomorphism classes: | 2 |
This isogeny class is simple and geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is ordinary.
| $p$-rank: | $2$ |
| Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $19$ | $551$ | $4636$ | $60059$ | $1036849$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $4$ | $30$ | $73$ | $234$ | $1014$ | $4071$ | $16216$ | $65874$ | $263737$ | $1047550$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+(x^3+a+1)y=(a+1)x^6+(a+1)x^4+x^3+ax+a$
- $y^2+(x^3+a)y=ax^6+ax^4+x^3+(a+1)x+a+1$
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 4.0.5225.1. |
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 |
|---|---|---|
| 2.4.b_h | $2$ | 2.16.n_cv |