Basic invariants
| Dimension: | $4$ |
| Group: | $S_5$ |
| Conductor: | \(173513\)\(\medspace = 167 \cdot 1039 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.5.173513.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_5$ |
| Parity: | even |
| Determinant: | 1.173513.2t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.5.173513.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - 2x^{4} - 5x^{3} + 3x^{2} + 3x - 1 \)
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The roots of $f$ are computed in $\Q_{ 293 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 134 + 23\cdot 293 + 118\cdot 293^{2} + 234\cdot 293^{3} + 53\cdot 293^{4} +O(293^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 148 + 12\cdot 293 + 22\cdot 293^{2} + 283\cdot 293^{3} + 171\cdot 293^{4} +O(293^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 154 + 150\cdot 293 + 141\cdot 293^{2} + 262\cdot 293^{3} + 195\cdot 293^{4} +O(293^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 218 + 7\cdot 293 + 255\cdot 293^{2} + 240\cdot 293^{3} + 260\cdot 293^{4} +O(293^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 227 + 98\cdot 293 + 49\cdot 293^{2} + 151\cdot 293^{3} + 196\cdot 293^{4} +O(293^{5})\)
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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.