Invariants
| Base field: | $\F_{2^{4}}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 7 x + 25 x^{2} - 112 x^{3} + 256 x^{4}$ |
| Frobenius angles: | $\pm0.0534491861630$, $\pm0.535384626806$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.669977.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $4$ |
| Isomorphism classes: | 4 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $163$ | $65363$ | $16159168$ | $4238725187$ | $1099211305483$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $10$ | $258$ | $3943$ | $64674$ | $1048290$ | $16779279$ | $268399834$ | $4294855170$ | $68719886839$ | $1099512784018$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+(x^3+(a+1)x+a+1)y=(a^3+a^2+a+1)x^6+ax^5+ax^4+(a^2+1)x^3+(a^2+a)x^2+(a+1)x+a+1$
- $y^2+(x^3+(a^2+1)x+a^2+1)y=(a^3+a+1)x^6+a^2x^5+a^2x^4+ax^3+(a^2+1)x^2+(a^2+1)x+a^2+a+1$
- $y^2+(x^3+ax+a)y=(a^3+a+1)x^6+(a+1)x^5+(a+1)x^4+a^2x^3+(a^3+a)x^2+ax+a^3+a^2+a$
- $y^2+(x^3+a^2x+a^2)y=(a^3+1)x^6+(a^2+1)x^5+(a^2+1)x^4+(a+1)x^3+a^3x^2+a^2x+a^3+a+1$
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 4.0.669977.1. |
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.h_z | $2$ | 2.256.b_aqp |