Invariants
| Base field: | $\F_{2^{2}}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + x^{2} + 16 x^{4}$ |
| Frobenius angles: | $\pm0.269946543837$, $\pm0.730053456163$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(i, \sqrt{7})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $6$ |
| Isomorphism classes: | 7 |
This isogeny class is simple but not 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $18$ | $324$ | $4050$ | $82944$ | $1049778$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $5$ | $19$ | $65$ | $319$ | $1025$ | $4003$ | $16385$ | $64639$ | $262145$ | $1050979$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+(x^2+x+a)y=(a+1)x^5+ax^3+x+a$
- $y^2+(x^2+x+a)y=(a+1)x^5+(a+1)x^4+ax^3+(a+1)x^2+x$
- $y^2+(x^2+x+a+1)y=ax^5+(a+1)x^3+x+a+1$
- $y^2+(x^2+x+a+1)y=ax^5+(a+1)x^4+(a+1)x^3+(a+1)x^2+x+a$
- $y^2+(x^2+x+a)y=ax^5+ax^3+x^2+a$
- $y^2+(x^2+x+a+1)y=(a+1)x^5+(a+1)x^3+x^2+a+1$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{4}}$.
Endomorphism algebra over $\F_{2^{2}}$| The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{7})\). |
| The base change of $A$ to $\F_{2^{4}}$ is 1.16.b 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$ |
Base change
This is a primitive isogeny class.