Basic invariants
| Dimension: | $4$ |
| Group: | $S_5$ |
| Conductor: | \(5913\)\(\medspace = 3^{4} \cdot 73 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.1.5913.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_5$ |
| Parity: | even |
| Determinant: | 1.73.2t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.1.5913.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - x^{4} + 2x^{3} - x^{2} + x + 1 \)
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The roots of $f$ are computed in $\Q_{ 457 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 45 + 171\cdot 457 + 240\cdot 457^{2} + 61\cdot 457^{3} + 205\cdot 457^{4} +O(457^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 102 + 353\cdot 457 + 118\cdot 457^{2} + 178\cdot 457^{3} + 93\cdot 457^{4} +O(457^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 149 + 343\cdot 457 + 127\cdot 457^{2} + 182\cdot 457^{3} + 368\cdot 457^{4} +O(457^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 237 + 251\cdot 457 + 79\cdot 457^{2} + 429\cdot 457^{3} + 235\cdot 457^{4} +O(457^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 382 + 251\cdot 457 + 347\cdot 457^{2} + 62\cdot 457^{3} + 11\cdot 457^{4} +O(457^{5})\)
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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 | Complex conjugation |
| $1$ | $1$ | $()$ | $4$ | |
| $10$ | $2$ | $(1,2)$ | $2$ | |
| $15$ | $2$ | $(1,2)(3,4)$ | $0$ | ✓ |
| $20$ | $3$ | $(1,2,3)$ | $1$ | |
| $30$ | $4$ | $(1,2,3,4)$ | $0$ | |
| $24$ | $5$ | $(1,2,3,4,5)$ | $-1$ | |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |