Invariants
Base field: | $\F_{2^{4}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 13 x + 73 x^{2} - 208 x^{3} + 256 x^{4}$ |
Frobenius angles: | $\pm0.0987587980325$, $\pm0.265114785720$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.5225.1 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $109$ | $60059$ | $16884100$ | $4317100979$ | $1100590570909$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $4$ | $234$ | $4123$ | $65874$ | $1049604$ | $16777263$ | $268424524$ | $4294949154$ | $68719785643$ | $1099514682154$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+(x^3+a^2+a+1)y=(a^3+a^2+a)x^6+(a^2+a+1)x^4+x^3+(a^2+a)x+a^2+1$
- $y^2+(x^3+a^2+a)y=a^3x^6+(a^2+a)x^4+x^3+(a^2+a+1)x+a^3+a+1$
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 4.0.5225.1. |
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.n_cv | $2$ | 2.256.ax_qr |