Invariants
| Base field: | $\F_{2^{4}}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 9 x + 43 x^{2} - 144 x^{3} + 256 x^{4}$ |
| Frobenius angles: | $\pm0.108303609292$, $\pm0.441636942625$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}, \sqrt{37})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $12$ |
| Isomorphism classes: | 12 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $147$ | $66591$ | $16780932$ | $4263222411$ | $1098088275837$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $8$ | $262$ | $4097$ | $65050$ | $1047218$ | $16784647$ | $268490636$ | $4295073394$ | $68719476737$ | $1099513109302$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 12 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+(x^3+ax+a)y=a^3x^6+(a^3+a^2)x^5+(a^3+a^2)x^4+(a^3+a+1)x^3+(a^2+a)x^2+(a^3+a+1)x+a^2+a$
- $y^2+(x^3+a^2x+a^2)y=(a^3+a+1)x^6+(a^3+a^2+a+1)x^5+(a^3+a^2+a+1)x^4+(a^3+1)x^3+(a^3+a^2+a)x^2+(a^3+1)x+a^3+a^2+a$
- $y^2+(x^3+a^2+a+1)y=(a^2+a+1)x^4+(a+1)x^3+(a^2+a)x+a^3+1$
- $y^2+(x^3+(a+1)x+a+1)y=(a^2+a+1)x^5+(a^2+a+1)x^4+(a^2+a+1)x^3+a^3x+1$
- $y^2+(x^3+(a+1)x+a+1)y=a^3x^6+(a^3+a)x^5+(a^3+a)x^4+(a^3+a^2+1)x^3+(a^3+a^2+a)x^2+(a^3+a^2+1)x+a^3+a^2+a$
- $y^2+(x^3+a^2+a)y=(a^2+a)x^4+(a^2+1)x^3+(a^2+a+1)x+a^3+a^2+1$
- $y^2+(x^3+ax+a)y=(a^3+a)x^6+(a^3+a^2+1)x^5+(a^3+a^2+1)x^4+(a^3+a^2+1)x^3+(a^3+a^2+a)x^2+x+a^2+a$
- $y^2+(x^3+(a^2+1)x+a^2+1)y=(a^3+1)x^6+a^3x^5+a^3x^4+(a^3+a^2+a)x^3+x^2+(a^3+a^2+a)x+1$
- $y^2+(x^3+a^2x+a^2)y=(a^2+a)x^5+(a^2+a)x^4+(a^2+a)x^3+(a^3+a)x+1$
- $y^2+(x^3+a^2+a)y=(a^2+a)x^4+a^2x^3+(a^2+a+1)x+a^3+a+1$
- $y^2+(x^3+a^2x+a^2)y=(a^3+a)x^6+(a^3+a^2+a)x^5+(a^3+a^2+a)x^4+(a^3+a^2+a)x^3+(a^3+a^2+1)x^2+x+1$
- $y^2+(x^3+a^2+a+1)y=(a^2+a+1)x^4+ax^3+(a^2+a)x+a^3+a^2+a$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{24}}$.
Endomorphism algebra over $\F_{2^{4}}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{37})\). |
| The base change of $A$ to $\F_{2^{24}}$ is 1.16777216.fmx 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-111}) \)$)$ |
- Endomorphism algebra over $\F_{2^{8}}$
The base change of $A$ to $\F_{2^{8}}$ is the simple isogeny class 2.256.f_aix and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{37})\). - Endomorphism algebra over $\F_{2^{12}}$
The base change of $A$ to $\F_{2^{12}}$ is the simple isogeny class 2.4096.a_fmx and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{37})\).
Base change
This is a primitive isogeny class.