Invariants
Base field: | $\F_{2^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 3 x + 9 x^{2} - 12 x^{3} + 16 x^{4}$ |
Frobenius angles: | $\pm0.272875599394$, $\pm0.469557725221$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.3625.1 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $11$ | $451$ | $5456$ | $65395$ | $1046771$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $2$ | $26$ | $83$ | $258$ | $1022$ | $4151$ | $16298$ | $64738$ | $261227$ | $1051106$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+(x^3+x+1)y=ax^5+ax^4+x^3+x+a+1$
- $y^2+(x^3+x+1)y=(a+1)x^5+(a+1)x^4+x^3+x+a$
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 4.0.3625.1. |
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.d_j | $2$ | 2.16.j_bp |