Invariants
| Base field: | $\F_{2}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 3 x + 5 x^{2} + 6 x^{3} + 4 x^{4}$ |
| Frobenius angles: | $\pm0.543118021706$, $\pm0.876451355039$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}, \sqrt{5})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $1$ |
| Isomorphism classes: | 1 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $19$ | $19$ | $76$ | $171$ | $1159$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $6$ | $6$ | $9$ | $10$ | $36$ | $87$ | $90$ | $274$ | $513$ | $1086$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:
- $y^2+(x^3+x+1)y=x^5+x^4+x^3+x$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{6}}$.
Endomorphism algebra over $\F_{2}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{5})\). |
| The base change of $A$ to $\F_{2^{6}}$ is 1.64.l 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$ |
- Endomorphism algebra over $\F_{2^{2}}$
The base change of $A$ to $\F_{2^{2}}$ is the simple isogeny class 2.4.b_ad and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{5})\). - Endomorphism algebra over $\F_{2^{3}}$
The base change of $A$ to $\F_{2^{3}}$ is the simple isogeny class 2.8.a_l and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{5})\).
Base change
This is a primitive isogeny class.