Invariants
| Base field: | $\F_{2^{2}}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 3 x + 5 x^{2} - 12 x^{3} + 16 x^{4}$ |
| Frobenius angles: | $\pm0.103279877171$, $\pm0.563386789496$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}, \sqrt{-7})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $2$ |
| Isomorphism classes: | 2 |
This isogeny class is simple but not geometrically simple, not 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $7$ | $259$ | $3136$ | $58275$ | $1109227$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $2$ | $18$ | $47$ | $226$ | $1082$ | $4191$ | $16298$ | $65986$ | $264143$ | $1049778$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+(x^3+a+1)y=ax^6+(a+1)x^4+(a+1)x^3+ax+1$
- $y^2+(x^3+a)y=(a+1)x^6+ax^4+ax^3+(a+1)x+1$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{6}}$.
Endomorphism algebra over $\F_{2^{2}}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-7})\). |
| The base change of $A$ to $\F_{2^{6}}$ is 1.64.aj 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$ |
Base change
This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{2}}$.
| Subfield | Primitive Model |
| $\F_{2}$ | 2.2.ab_ab |
| $\F_{2}$ | 2.2.b_ab |