Basic invariants
| Dimension: | $4$ |
| Group: | $S_5$ |
| Conductor: | \(68571\)\(\medspace = 3^{2} \cdot 19 \cdot 401 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.3.68571.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_5$ |
| Parity: | odd |
| Determinant: | 1.7619.2t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.3.68571.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - x^{4} - x^{3} + x^{2} - 5x - 1 \)
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The roots of $f$ are computed in $\Q_{ 149 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 20 + 21\cdot 149 + 96\cdot 149^{2} + 19\cdot 149^{3} + 94\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 61 + 137\cdot 149 + 94\cdot 149^{2} + 119\cdot 149^{3} + 15\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 80 + 44\cdot 149 + 42\cdot 149^{2} + 98\cdot 149^{3} + 57\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 143 + 11\cdot 149 + 129\cdot 149^{2} + 79\cdot 149^{3} + 89\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 144 + 82\cdot 149 + 84\cdot 149^{2} + 129\cdot 149^{3} + 40\cdot 149^{4} +O(149^{5})\)
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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 |
| $1$ | $1$ | $()$ | $4$ |
| $10$ | $2$ | $(1,2)$ | $2$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $20$ | $3$ | $(1,2,3)$ | $1$ |
| $30$ | $4$ | $(1,2,3,4)$ | $0$ |
| $24$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.