Invariants
| Base field: | $\F_{2^{6}}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 27 x + 307 x^{2} - 1728 x^{3} + 4096 x^{4}$ |
| Frobenius angles: | $\pm0.0943151045543$, $\pm0.239018228779$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}, \sqrt{13})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $14$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $2649$ | $16315191$ | $68719584492$ | $281559424997691$ | $1152972791571810999$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $38$ | $3982$ | $262145$ | $16782250$ | $1073789588$ | $68719692247$ | $4398046363658$ | $281474968487314$ | $18014398509481985$ | $1152921505993895902$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 14 curves (of which all are hyperelliptic):
- $y^2+(x^3+(a^4+a^2) x+a^4+a^2) y=(a^5+a^4+a^3+a+1) x^6+(a^5+1) x^5+(a^5+1) x^4+(a^5+1) x^3+(a^4+a^3+a^2+a+1) x^2+(a^4+a^3+a+1) x+a^4+a+1$
- $y^2+(x^3+a^2 x+a^2) y=(a^3+a^2+a) x^6+(a^5+a^4+a^3) x^5+(a^5+a^4+a^3) x^4+(a^5+a^4+a^3) x^3+(a^3+a^2+1) x^2+(a^5+a^4+a^2) x+a^5+a^4+a^2$
- $y^2+(x^3+(a^5+a^2) x+a^5+a^2) y=(a^5+a^4+a^2) x^5+(a^5+a^4+a^2) x^4+(a^5+a^2+a+1) x^3+(a^4+a^3+a^2+a+1) x+a^4$
- $y^2+(x^3+(a^5+1) x+a^5+1) y=(a^4+a^2+a) x^5+(a^4+a^2+a) x^4+(a^5+a^2) x^3+(a^5+a^4+a^3+1) x+a^5+a^4+a^2+a+1$
- $y^2+(x^3+(a^4+a+1) x+a^4+a+1) y=(a^4+a^2+1) x^5+(a^4+a^2+1) x^4+(a^3+a+1) x^3+(a^5+a^3+a^2+1) x+a^5+a^3+a^2+a$
- $y^2+(x^3+(a^5+a^4) x+a^5+a^4) y=(a^5+a+1) x^5+(a^5+a+1) x^4+(a^5+1) x^3+(a^4+a^3+a^2) x+a^4+a+1$
- $y^2+(x^3+(a^5+a^4+a) x+a^5+a^4+a) y=(a^5+a^2+a) x^5+(a^5+a^2+a) x^4+(a^4+a^3+a^2+a) x^3+(a^5+a^4+a^3+a+1) x+a^5+a^4+a^3+a^2+a$
- $y^2+(x^3+(a^2+a) x+a^2+a) y=(a^5+a^4+a^2) x^5+(a^5+a^4+a^2) x^4+(a^5+a^4) x^3+(a^5+a^4+a^3+a^2) x+a^5+a^4+a$
- $y^2+(x^3+a x+a) y=(a^5+a^2+1) x^5+(a^5+a^2+1) x^4+(a^5+a^4+a^3) x^3+(a^4+a^3) x+a^3+1$
- $y^2+(x^3+(a^5+a^2+a+1) x+a^5+a^2+a+1) y=(a^5+a+1) x^5+(a^5+a+1) x^4+(a^4+a^2) x^3+(a^3+a) x+a^2$
- $y^2+(x^3+(a^5+a^4+a^2+a+1) x+a^5+a^4+a^2+a+1) y=(a^2+a+1) x^5+(a^2+a+1) x^4+(a^3+a^2) x^3+(a^4+a^3+1) x+a^5+a^3+a$
- $y^2+(x^3+(a^4+a^2) x+a^4+a^2) y=(a^4+a^2+a) x^5+(a^4+a^2+a) x^4+(a^2+a) x^3+(a^3+a^2+1) x+a$
- $y^2+(x^3+a^4 x+a^4) y=(a^5+a^4+1) x^5+(a^5+a^4+1) x^4+(a^5+a^4+a^3+a^2+1) x^3+(a^5+a^4+a^3+a) x+a^5+a^3$
- $y^2+(x^3+a^2 x+a^2) y=a^5 x^5+a^5 x^4+(a^4+a^3+a^2+1) x^3+(a^5+a^3+a^2) x+a^4+a^3+a$
where $a$ is a root of the Conway polynomial.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{36}}$.
Endomorphism algebra over $\F_{2^{6}}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{13})\). |
| The base change of $A$ to $\F_{2^{36}}$ is 1.68719476736.gdkl 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-39}) \)$)$ |
- Endomorphism algebra over $\F_{2^{12}}$
The base change of $A$ to $\F_{2^{12}}$ is the simple isogeny class 2.4096.ael_nnd and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{13})\). - Endomorphism algebra over $\F_{2^{18}}$
The base change of $A$ to $\F_{2^{18}}$ is the simple isogeny class 2.262144.a_gdkl and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{13})\).
Base change
This is a primitive isogeny class.