Invariants
Base field: | $\F_{2^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - x - 4 x^{3} + 16 x^{4}$ |
Frobenius angles: | $\pm0.180745703069$, $\pm0.702084401492$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.13068.1 |
Galois group: | $D_{4}$ |
Jacobians: | $2$ |
This isogeny class is simple and geometrically simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
$p$-rank: | $1$ |
Slopes: | $[0, 1/2, 1/2, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $12$ | $264$ | $3348$ | $78672$ | $1110972$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $4$ | $16$ | $52$ | $304$ | $1084$ | $4120$ | $16804$ | $65440$ | $261196$ | $1048936$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+xy=x^5+ax^3+x$
- $y^2+xy=x^5+(a+1)x^3+x$
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 4.0.13068.1. |
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_a | $2$ | 2.16.ab_y |