Invariants
Base field: | $\F_{2}$ |
Dimension: | $3$ |
L-polynomial: | $1 - 2 x + x^{2} + 2 x^{4} - 8 x^{5} + 8 x^{6}$ |
Frobenius angles: | $\pm0.0693533547550$, $\pm0.339907131295$, $\pm0.770553776540$ |
Angle rank: | $2$ (numerical) |
Number field: | 6.0.2580992.1 |
Galois group: | $D_{6}$ |
Jacobians: | $1$ |
Isomorphism classes: | 1 |
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: | $2$ |
Slopes: | $[0, 0, 1/2, 1/2, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $2$ | $44$ | $386$ | $5984$ | $16742$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $1$ | $3$ | $7$ | $23$ | $11$ | $51$ | $127$ | $255$ | $619$ | $1043$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is not hyperelliptic), and hence is principally polarizable:
- $x^4+x^3z+xy^3+xz^3+y^4=0$
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 6.0.2580992.1. |
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 |
---|---|---|
3.2.c_b_a | $2$ | 3.4.ac_f_am |