Invariants
| Base field: | $\F_{2^{4}}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 7 x + 37 x^{2} - 112 x^{3} + 256 x^{4}$ |
| Frobenius angles: | $\pm0.218216417042$, $\pm0.467819082007$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.54665.2 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $175$ | $72275$ | $17189200$ | $4294941875$ | $1100532041875$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $10$ | $282$ | $4195$ | $65538$ | $1049550$ | $16788567$ | $268453930$ | $4294785378$ | $68718541915$ | $1099510978402$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 24 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+(x^3+x+1)y=(a^3+a)x^5+(a^3+a)x^4+(a^2+1)x^3+(a^2+1)x+a^3+a^2+a+1$
- $y^2+(x^3+x+1)y=(a^3+a^2+a+1)x^5+(a^3+a^2+a+1)x^4+a^2x^3+a^2x+a^3+a+1$
- $y^2+(x^3+ax+a)y=(a^3+a^2+a+1)x^6+ax^5+ax^4+a^2x^3+(a^3+1)x^2+a^2x+a^3+1$
- $y^2+(x^3+(a+1)x+a+1)y=(a^3+a^2+1)x^6+(a^3+a)x^5+(a^3+a)x^4+(a^3+a^2)x^3+(a^3+a^2)x^2+(a^3+a^2+a)x+a^3+a^2+a+1$
- $y^2+(x^3+x+1)y=(a^3+a+1)x^5+(a^3+a+1)x^4+(a+1)x^3+(a+1)x+a^3$
- $y^2+(x^3+x+1)y=(a^3+a)x^5+(a^3+a)x^4+(a+1)x^3+(a+1)x+a^3+1$
- $y^2+(x^3+x+1)y=a^3x^5+a^3x^4+ax^3+ax+a^3+a$
- $y^2+(x^3+a^2x+a^2)y=(a^3+1)x^6+a^2x^5+a^2x^4+(a+1)x^3+a^3x^2+(a+1)x+a^3$
- $y^2+(x^3+(a^2+1)x+a^2+1)y=(a^3+a+1)x^6+a^3x^5+a^3x^4+(a^3+a^2+a+1)x^3+(a^2+1)x^2+(a^3+a+1)x$
- $y^2+(x^3+x+1)y=(a^3+1)x^5+(a^3+1)x^4+(a^2+1)x^3+(a^2+1)x+a^3+a^2$
- $y^2+(x^3+ax+a)y=(a^3+a^2+1)x^6+(a^3+a^2)x^5+(a^3+a^2)x^4+(a^3+a)x^3+x^2+(a^3+1)x+a+1$
- $y^2+(x^3+x+1)y=a^3x^5+a^3x^4+(a^2+1)x^3+(a^2+1)x+a^3+a^2+1$
- $y^2+(x^3+x+1)y=(a^3+a^2+1)x^5+(a^3+a^2+1)x^4+ax^3+ax+a^3+a^2+a+1$
- $y^2+(x^3+x+1)y=(a^3+a^2)x^5+(a^3+a^2)x^4+a^2x^3+a^2x+a^3$
- $y^2+(x^3+(a+1)x+a+1)y=(a^3+a^2)x^6+(a+1)x^5+(a+1)x^4+(a^2+1)x^3+(a^3+1)x^2+(a^2+1)x+a^3+1$
- $y^2+(x^3+x+1)y=a^3x^5+a^3x^4+(a+1)x^3+(a+1)x+a^3+a+1$
- $y^2+(x^3+a^2x+a^2)y=(a^3+a^2+1)x^6+(a^3+a^2+a+1)x^5+(a^3+a^2+a+1)x^4+a^3x^3+a^2x^2+(a^3+a^2+1)x$
- $y^2+(x^3+x+1)y=(a^3+a^2+a)x^5+(a^3+a^2+a)x^4+a^2x^3+a^2x+a^3+a$
- $y^2+(x^3+x+1)y=(a^3+a^2+a+1)x^5+(a^3+a^2+a+1)x^4+(a+1)x^3+(a+1)x+a^3+a^2$
- $y^2+(x^3+x+1)y=(a^3+a^2)x^5+(a^3+a^2)x^4+ax^3+ax+a^3+a^2+a$
- $y^2+(x^3+(a^2+1)x+a^2+1)y=(a^3+a)x^6+(a^2+1)x^5+(a^2+1)x^4+ax^3+(a^2+a+1)x^2+ax+a^2+a+1$
- $y^2+(x^3+x+1)y=(a^3+a)x^5+(a^3+a)x^4+a^2x^3+a^2x+a^3+a^2+a$
- $y^2+(x^3+x+1)y=(a^3+a^2+a+1)x^5+(a^3+a^2+a+1)x^4+ax^3+ax+a^3+a^2+1$
- $y^2+(x^3+x+1)y=(a^3+a^2)x^5+(a^3+a^2)x^4+(a^2+1)x^3+(a^2+1)x+a^3+1$
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 4.0.54665.2. |
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.h_bl | $2$ | 2.256.z_mb |