Basic invariants
Defining polynomial
| $f(x)$ | $=$ | \(x^{5} - x^{4} - 24 x^{3} + 17 x^{2} + 41 x + 13\)  . |
The roots of $f$ are computed in $\Q_{ 11 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ | \( 4 + 8\cdot 11 + 8\cdot 11^{2} + 2\cdot 11^{3} + 3\cdot 11^{4} +O(11^{5})\)
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| $r_{ 2 }$ | $=$ | \( 5 + 10\cdot 11 + 8\cdot 11^{2} + 5\cdot 11^{3} + 6\cdot 11^{4} +O(11^{5})\)
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| $r_{ 3 }$ | $=$ | \( 6 + 6\cdot 11 + 7\cdot 11^{2} + 3\cdot 11^{3} +O(11^{5})\)
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| $r_{ 4 }$ | $=$ | \( 9 + 6\cdot 11 + 5\cdot 11^{2} + 6\cdot 11^{3} + 8\cdot 11^{4} +O(11^{5})\)
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| $r_{ 5 }$ | $=$ | \( 10 + 2\cdot 11^{2} + 3\cdot 11^{3} + 3\cdot 11^{4} +O(11^{5})\)
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Generators of the action on the roots $r_1, \ldots, r_{ 5 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 5 }$ | Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $5$ | $(1,4,5,2,3)$ | $\zeta_{5}^{3}$ |
| $1$ | $5$ | $(1,5,3,4,2)$ | $\zeta_{5}$ |
| $1$ | $5$ | $(1,2,4,3,5)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ |
| $1$ | $5$ | $(1,3,2,5,4)$ | $\zeta_{5}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.