Basic invariants
Defining polynomial
| $f(x)$ | $=$ | \(x^{3} - x^{2} - 4 x - 1\)  . |
The roots of $f$ are computed in $\Q_{ 5 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ | \( 1 + 3\cdot 5 + 2\cdot 5^{2} + 5^{3} + 3\cdot 5^{4} +O(5^{5})\)
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| $r_{ 2 }$ | $=$ | \( 2 + 4\cdot 5 + 3\cdot 5^{2} + 3\cdot 5^{3} + 2\cdot 5^{4} +O(5^{5})\)
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| $r_{ 3 }$ | $=$ | \( 3 + 2\cdot 5 + 3\cdot 5^{2} + 4\cdot 5^{3} + 3\cdot 5^{4} +O(5^{5})\)
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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.