Basic invariants
Defining polynomial
$f(x)$ | $=$ | \(x^{3} - x^{2} - 2 x + 1\) ![]() |
The roots of $f$ are computed in $\Q_{ 13 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 3 + 2\cdot 13 + 5\cdot 13^{2} + 5\cdot 13^{3} + 11\cdot 13^{4} +O(13^{5})\) ![]() |
$r_{ 2 }$ | $=$ | \( 5 + 10\cdot 13 + 3\cdot 13^{2} + 8\cdot 13^{3} + 12\cdot 13^{4} +O(13^{5})\) ![]() |
$r_{ 3 }$ | $=$ | \( 6 + 4\cdot 13^{2} + 12\cdot 13^{3} + 13^{4} +O(13^{5})\) ![]() |
Generators of the action on the roots $ r_{ 1 }, r_{ 2 }, r_{ 3 } $
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $ r_{ 1 }, r_{ 2 }, r_{ 3 } $ | Character value |
$1$ | $1$ | $()$ | $1$ |
$1$ | $3$ | $(1,2,3)$ | $-\zeta_{3} - 1$ |
$1$ | $3$ | $(1,3,2)$ | $\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.