Invariants
| Base field: | $\F_{2^{3}}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 5 x + 17 x^{2} - 40 x^{3} + 64 x^{4}$ |
| Frobenius angles: | $\pm0.178413517577$, $\pm0.488253149089$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}, \sqrt{-7})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $6$ |
| Isomorphism classes: | 4 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $37$ | $4699$ | $268324$ | $16582771$ | $1082796157$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $4$ | $74$ | $523$ | $4050$ | $33044$ | $264143$ | $2099948$ | $16771234$ | $134202619$ | $1073751914$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+(x^3+(a+1)x+a+1)y=x^6+a^2x^5+a^2x^4+(a^2+1)x^2+(a^2+1)x+a^2+a+1$
- $y^2+(x^3+(a^2+1)x+a^2+1)y=x^6+(a^2+a)x^5+(a^2+a)x^4+(a^2+a+1)x^2+(a^2+a+1)x+a+1$
- $y^2+(x^3+(a^2+a+1)x+a^2+a+1)y=x^6+ax^5+ax^4+(a+1)x^2+(a+1)x+a^2+1$
- $y^2+(x^3+(a^2+1)x+a^2+1)y=(a^2+1)x^5+(a^2+1)x^4+ax^3+a^2+a+1$
- $y^2+(x^3+(a+1)x+a+1)y=(a+1)x^5+(a+1)x^4+(a^2+a)x^3+a^2+1$
- $y^2+(x^3+(a^2+a+1)x+a^2+a+1)y=(a^2+a+1)x^5+(a^2+a+1)x^4+a^2x^3+a+1$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{9}}$.
Endomorphism algebra over $\F_{2^{3}}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-7})\). |
| The base change of $A$ to $\F_{2^{9}}$ is 1.512.f 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$ |
Base change
This is a primitive isogeny class.