Invariants
Base field: | $\F_{2^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - x - x^{2} - 4 x^{3} + 16 x^{4}$ |
Frobenius angles: | $\pm0.153921966489$, $\pm0.719142225765$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.45177.1 |
Galois group: | $D_{4}$ |
Jacobians: | $2$ |
Isomorphism classes: | 2 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $11$ | $231$ | $3212$ | $76923$ | $1079441$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $4$ | $14$ | $49$ | $298$ | $1054$ | $4151$ | $16888$ | $65554$ | $262417$ | $1050014$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+(x^3+a+1)y=(a+1)x^4+ax^3+ax+1$
- $y^2+(x^3+a)y=ax^4+(a+1)x^3+(a+1)x+1$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{2}}$.
Endomorphism algebra over $\F_{2^{2}}$The endomorphism algebra of this simple isogeny class is 4.0.45177.1. |
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 |
---|---|---|
2.4.b_ab | $2$ | 2.16.ad_z |