Invariants
Base field: | $\F_{2}$ |
Dimension: | $2$ |
L-polynomial: | $1 - x + x^{2} - 2 x^{3} + 4 x^{4}$ |
Frobenius angles: | $\pm0.197201053961$, $\pm0.652365995579$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.2873.1 |
Galois group: | $D_{4}$ |
Jacobians: | $1$ |
Isomorphism classes: | 1 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $3$ | $27$ | $36$ | $459$ | $1803$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $2$ | $6$ | $5$ | $26$ | $52$ | $63$ | $142$ | $274$ | $437$ | $966$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:
- $y^2+(x^3+x+1)y=x^3+x+1$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2}$.
Endomorphism algebra over $\F_{2}$The endomorphism algebra of this simple isogeny class is 4.0.2873.1. |
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 |
---|---|---|
2.2.b_b | $2$ | 2.4.b_f |