Basic invariants
| Dimension: | $4$ |
| Group: | $S_5$ |
| Conductor: | \(220036\)\(\medspace = 2^{2} \cdot 55009 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.5.220036.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_5$ |
| Parity: | even |
| Determinant: | 1.55009.2t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.5.220036.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - 2x^{4} - 5x^{3} + 11x^{2} - 2x - 1 \)
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The roots of $f$ are computed in $\Q_{ 83 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 12 + 38\cdot 83 + 44\cdot 83^{2} + 73\cdot 83^{3} + 46\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 37 + 9\cdot 83 + 43\cdot 83^{2} + 33\cdot 83^{3} + 35\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 45 + 72\cdot 83 + 68\cdot 83^{2} + 58\cdot 83^{3} + 48\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 77 + 72\cdot 83 + 42\cdot 83^{2} + 51\cdot 83^{3} + 52\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 80 + 55\cdot 83 + 49\cdot 83^{2} + 31\cdot 83^{3} + 65\cdot 83^{4} +O(83^{5})\)
|
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.