Invariants
| Base field: | $\F_{2^{4}}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 8 x + 41 x^{2} - 128 x^{3} + 256 x^{4}$ |
| Frobenius angles: | $\pm0.187929673264$, $\pm0.445855430413$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.107408.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $24$ |
| Isomorphism classes: | 24 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $162$ | $70308$ | $17147538$ | $4292725248$ | $1100005557282$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $9$ | $275$ | $4185$ | $65503$ | $1049049$ | $16789043$ | $268483833$ | $4294938559$ | $68718656313$ | $1099509407315$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 24 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+(x^2+x+a^3+a^2+a)y=(a^3+a^2+a)x^5+(a^3+a^2+1)x^4+(a^3+a^2+1)x^3+(a^3+a^2+1)x^2+(a^3+a)x+a^2+a+1$
- $y^2+(x^2+x+a^3+a+1)y=(a^3+a+1)x^5+(a^3+a)x^4+(a^3+a^2+a)x^3+(a^3+a)x^2+a^3x+a^2$
- $y^2+(x^2+x+a^3+1)y=(a^3+1)x^5+(a^3+a^2+1)x^4+(a^3+a+1)x^3+(a^3+a^2+1)x^2+(a^3+a^2)x+a^3+a^2+a+1$
- $y^2+(x^2+x+a^3+a^2+1)y=(a^3+a^2+1)x^5+(a^3+1)x^4+(a^3+1)x^3+(a^3+1)x^2+(a^3+a^2+a+1)x+a^2+a$
- $y^2+(x^2+x+a^3+1)y=(a^3+a)x^5+(a^3+a^2)x^3+(a^3+a^2+a)x+a+1$
- $y^2+(x^2+x+a^3+a^2)y=(a^3+a^2+a+1)x^5+(a^3+a^2+a)x^3+(a^2+a+1)x+a^3+a^2$
- $y^2+(x^2+x+a^3+a^2+a)y=(a^3+a+1)x^5+(a^3+a^2+a)x^4+(a^3+a)x^3+(a^3+a^2+a)x^2+(a^2+a)x+a^2+a$
- $y^2+(x^2+x+a^3)y=(a^3+a+1)x^5+(a^3+a^2+1)x^3+(a^3+a^2+a)x+a^2+1$
- $y^2+(x^2+x+a^3+a^2+a+1)y=(a^3+a^2+1)x^5+(a^3+a+1)x^3+(a^3+1)x+a^2$
- $y^2+(x^2+x+a^3+a+1)y=(a^3+1)x^5+(a^3+a^2+1)x^4+a^3x^3+(a^3+a^2+1)x^2+(a^2+a+1)x+a^2$
- $y^2+(x^2+x+a^3+a^2+a+1)y=(a^3+a)x^5+(a^3+a+1)x^3+(a^2+a)x+a^3+a^2+a+1$
- $y^2+(x^2+x+a^3+a^2+1)y=a^3x^5+(a^3+a^2+a+1)x^3+(a^3+a+1)x+a^2+1$
- $y^2+(x^2+x+a^3+a^2+1)y=(a^3+a+1)x^5+(a^3+1)x^3+(a+1)x+a^3+1$
- $y^2+(x^2+x+a^3+a+1)y=(a^3+a^2+a+1)x^5+a^3x^3+(a^3+a^2+1)x+a^2$
- $y^2+(x^2+x+a^3)y=(a^3+a^2)x^5+(a^3+a^2+1)x^3+(a^2+a)x+a^3$
- $y^2+(x^2+x+a^3+a^2+1)y=(a^3+a^2+a)x^5+(a^3+a^2)x^4+(a^3+a^2+a+1)x^3+(a^3+a^2)x^2+(a^2+a+1)x+a^3+1$
- $y^2+(x^2+x+a^3+a^2)y=(a^3+1)x^5+(a^3+a^2+a)x^3+(a^3+a+1)x+a$
- $y^2+(x^2+x+a^3+a)y=(a^3+a^2+a)x^5+(a^3+1)x^3+(a^3+a^2+1)x+a+1$
- $y^2+(x^2+x+a^3+1)y=(a^3+a^2+a)x^5+(a^3+a+1)x^3+a^2x+a^3+a+1$
- $y^2+(x^2+x+a^3+a^2+a)y=(a^3+a^2)x^5+(a^3+a)x^3+(a^3+1)x+a$
- $y^2+(x^2+x+a^3+a)y=a^3x^5+(a^3+1)x^3+(a^2+a+1)x+a^3+a$
- $y^2+(x^2+x+a^3+1)y=(a^3+a^2+1)x^5+(a^3+1)x^4+(a^3+a^2)x^3+(a^3+1)x^2+(a^2+a)x+a^2+a$
- $y^2+(x^2+x+a^3+a+1)y=(a^3+a^2+1)x^5+(a^3+a^2+a)x^3+ax+a^3+a^2+a$
- $y^2+(x^2+x+a^3+a^2+a)y=(a^3+1)x^5+(a^3+a^2+1)x^3+(a^2+1)x+a^3+a^2+1$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{4}}$.
Endomorphism algebra over $\F_{2^{4}}$| The endomorphism algebra of this simple isogeny class is 4.0.107408.1. |
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 |
|---|---|---|
| 2.16.i_bp | $2$ | 2.256.s_fp |