Invariants
| Base field: | $\F_{2^{4}}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 8 x + 35 x^{2} - 128 x^{3} + 256 x^{4}$ |
| Frobenius angles: | $\pm0.100372839389$, $\pm0.484299017784$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.66417.2 |
| Galois group: | $D_{4}$ |
| Jacobians: | $16$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $156$ | $66768$ | $16554564$ | $4252053312$ | $1099002013356$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $9$ | $263$ | $4041$ | $64879$ | $1048089$ | $16787639$ | $268466025$ | $4294953055$ | $68719751865$ | $1099515571943$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 16 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+(x^2+x+a^3+a^2+a+1)y=(a+1)x^5+a^3x^4+ax^3+a^3x^2+(a+1)x$
- $y^2+(x^2+x+a^3+a)y=(a^2+1)x^5+(a^3+1)x^4+a^2x^3+(a^3+1)x^2+(a^2+1)x+a^3+a^2+a$
- $y^2+(x^2+x+a^3)y=ax^5+(a^3+a^2+1)x^4+(a+1)x^3+(a^3+a^2+1)x^2+ax+a^3+a+1$
- $y^2+(x^2+x+a^3+a+1)y=(a^2+a)x^5+(a^3+1)x^4+(a^2+1)x^3+(a^3+1)x^2+(a^3+a^2+a)x+a+1$
- $y^2+(x^2+x+a^3+a^2+1)y=(a^2+a)x^5+a^3x^4+a^2x^3+a^3x^2+(a^3+1)x$
- $y^2+(x^2+x)y=a^3x^5+(a^3+1)x^3+(a^3+a^2+1)x^2+(a^3+a^2)x$
- $y^2+(x^2+x+a^3)y=x^5+(a^2+1)x^3+(a^3+a^2+a)x+a^2+a$
- $y^2+(x^2+x+a^3+a^2)y=a^2x^5+(a^3+1)x^4+(a^2+1)x^3+(a^3+1)x^2+a^2x+a$
- $y^2+(x^2+x+a^3+a^2+a)y=(a^2+a+1)x^5+(a^3+a^2+a)x^4+(a+1)x^3+(a^3+a^2+a)x^2+(a^3+a^2+1)x+a^2+1$
- $y^2+(x^2+x+a^3+1)y=(a^2+a+1)x^5+(a^3+a+1)x^4+ax^3+(a^3+a+1)x^2+(a^3+a+1)x+a^3+a^2+1$
- $y^2+(x^2+x+a^3+a)y=x^5+(a+1)x^3+(a^3+a^2+1)x+a^2+a+1$
- $y^2+(x^2+x)y=a^3x^5+a^3x^4+(a^3+a+1)x^3+x^2+(a^3+a)x$
- $y^2+(x^2+x+a^3+a^2+a+1)y=x^5+a^2x^3+(a^3+1)x+a^2+a$
- $y^2+(x^2+x)y=(a^3+a+1)x^5+(a^3+a+1)x^4+(a^3+a^2+a)x^3+a^2x^2+(a^3+a)x$
- $y^2+(x^2+x+a^3+a^2)y=x^5+ax^3+(a^3+a+1)x+a^2+a+1$
- $y^2+(x^2+x)y=(a^3+a^2)x^5+(a^3+a^2+a+1)x^4+(a^3+a^2+1)x^3+(a^3+a^2+a)x$
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 4.0.66417.2. |
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.i_bj | $2$ | 2.256.g_alz |