Invariants
Base field: | $\F_{2}$ |
Dimension: | $3$ |
L-polynomial: | $1 - 2 x + x^{2} + x^{3} + 2 x^{4} - 8 x^{5} + 8 x^{6}$ |
Frobenius angles: | $\pm0.132091856901$, $\pm0.309487084859$, $\pm0.780459932197$ |
Angle rank: | $3$ (numerical) |
Number field: | 6.0.2369943.1 |
Galois group: | $S_4\times C_2$ |
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: | $3$ |
Slopes: | $[0, 0, 0, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $3$ | $63$ | $657$ | $9891$ | $26763$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $1$ | $3$ | $10$ | $31$ | $26$ | $72$ | $134$ | $263$ | $631$ | $1068$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is not hyperelliptic), and hence is principally polarizable:
- $x^4+x^3y+x^3z+xz^3+y^4+y^3z+y^2z^2=0$
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 6.0.2369943.1. |
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_ab | $2$ | 3.4.ac_j_an |