Invariants
Base field: | $\F_{2}$ |
Dimension: | $3$ |
L-polynomial: | $1 - 2 x + 2 x^{2} - x^{3} + 4 x^{4} - 8 x^{5} + 8 x^{6}$ |
Frobenius angles: | $\pm0.161334789180$, $\pm0.327009058845$, $\pm0.739882802642$ |
Angle rank: | $3$ (numerical) |
Number field: | 6.0.2464727.1 |
Galois group: | $S_4\times C_2$ |
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: | $3$ |
Slopes: | $[0, 0, 0, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $4$ | $104$ | $592$ | $11024$ | $31604$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $1$ | $5$ | $10$ | $33$ | $31$ | $62$ | $155$ | $257$ | $586$ | $1065$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which 1 is hyperelliptic), and hence is principally polarizable:
- $y^2+(x^4+x^2+x+1)y=x^8+x^7+x^5+x^3+x^2+1$
- $x^4+x^3y+x^2z^2+xz^3+y^4+y^3z+y^2z^2=0$
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 6.0.2464727.1. |
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_c_b | $2$ | 3.4.a_i_ab |