Invariants
| Base field: | $\F_{2^{6}}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 25 x + 279 x^{2} - 1600 x^{3} + 4096 x^{4}$ |
| Frobenius angles: | $\pm0.124520053775$, $\pm0.279743569050$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.2491209.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $12$ |
| Isomorphism classes: | 12 |
This isogeny class is simple and 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $2751$ | $16508751$ | $68850718524$ | $281602130861499$ | $1152969959859007101$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $40$ | $4030$ | $262645$ | $16784794$ | $1073786950$ | $68719565311$ | $4398046210780$ | $281474985654994$ | $18014398773767485$ | $1152921507875195350$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 12 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+(x^3+(a^2+a)x+a^2+a)y=(a^4+a^3+a)x^6+(a^5+a^4+a^3)x^5+(a^5+a^4+a^3)x^4+(a^4+a^3+a^2+a+1)x^3+(a^4+a)x^2+(a^2+a)x+a^5+a^4+a$
- $y^2+(x^3+(a^4+a^2)x+a^4+a^2)y=(a^5+a^4+a^3)x^6+(a^4+a^3+a^2+1)x^5+(a^4+a^3+a^2+1)x^4+(a^5+a^4+a^3+1)x^3+(a^5+a^3+a^2+1)x^2+(a^4+a^2)x+a^4+a^3+a^2$
- $y^2+(x^3+(a^5+a^2+a+1)x+a^5+a^2+a+1)y=(a^5+a^4+a^3+a^2)x^6+(a^5+a^4+a^3+a^2+1)x^5+(a^5+a^4+a^3+a^2+1)x^4+(a^4+a^3+a^2)x^3+(a^4+a^3)x^2+(a^5+a^2+a+1)x+a^4+a^3+a^2+a+1$
- $y^2+(x^3+(a^5+a^2)x+a^5+a^2)y=(a^4+a^3+a^2)x^6+(a^3+a^2)x^5+(a^3+a^2)x^4+(a^5+a^4+a^3+a^2)x^3+(a^5+a^3)x^2+(a^5+a^2)x+a^5+a^4+a^3+a^2+1$
- $y^2+(x^3+(a^5+a^2+a+1)x+a^5+a^2+a+1)y=(a^4+a^3+a^2+a+1)x^6+(a^4+a^3+a^2+1)x^5+(a^4+a^3+a^2+1)x^4+(a^4+a^3+a^2+a+1)x^3+(a^5+a^3)x^2+(a^5+a^3+a^2+a)x+a^4+a^3+a$
- $y^2+(x^3+(a^5+1)x+a^5+1)y=(a^5+a^3+a)x^6+(a^3+a+1)x^5+(a^3+a+1)x^4+(a^3+a^2+1)x^3+(a^5+a^4+a^3+a^2+a)x^2+(a^5+1)x+a^4+a^3$
- $y^2+(x^3+(a^5+a^2)x+a^5+a^2)y=(a^5+a^4+a^3+a^2+a)x^6+(a^5+a^4+a^3+a^2+1)x^5+(a^5+a^4+a^3+a^2+1)x^4+(a^5+a^4+a^3+1)x^3+a^5x^2+(a^5+a^4+a^3+a^2+a)x+a^5+a$
- $y^2+(x^3+(a^5+a^4)x+a^5+a^4)y=(a^4+a^3)x^6+(a^3+a+1)x^5+(a^3+a+1)x^4+(a^5+a^4+a^3+a^2)x^3+(a^3+a+1)x^2+(a^4+a^3+a)x+a^4+a^3+a+1$
- $y^2+(x^3+(a^5+a^4)x+a^5+a^4)y=(a^3+a^2+a)x^6+(a^4+a^3+a^2+a)x^5+(a^4+a^3+a^2+a)x^4+(a^3+a)x^3+(a^5+a^2)x^2+(a^5+a^4)x+1$
- $y^2+(x^3+(a^4+a^2)x+a^4+a^2)y=(a^4+a^3+a^2+a+1)x^6+(a^5+a^4+a^3)x^5+(a^5+a^4+a^3)x^4+(a^3+a)x^3+(a^3+1)x^2+(a^5+a^3+a)x+a^4+a^3+a$
- $y^2+(x^3+(a^5+1)x+a^5+1)y=(a^5+a^4+a^3+a)x^6+(a^3+a^2)x^5+(a^3+a^2)x^4+(a^4+a^3+a^2)x^3+(a^5+a^4+a^3+a^2)x^2+(a^3+1)x+a^5+a^4+a^3$
- $y^2+(x^3+(a^2+a)x+a^2+a)y=(a^4+a^3)x^6+(a^4+a^3+a^2+a)x^5+(a^4+a^3+a^2+a)x^4+(a^3+a^2+1)x^3+(a^5+a^4+a^3+a)x^2+(a^5+a^3)x+a^3+a^2+1$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{6}}$.
Endomorphism algebra over $\F_{2^{6}}$| The endomorphism algebra of this simple isogeny class is 4.0.2491209.1. |
Base change
This is a primitive isogeny class.
Twists
Below is a list of all twists of this isogeny class.
| Twist | Extension degree | Common base change |
|---|---|---|
| 2.64.z_kt | $2$ | (not in LMFDB) |