Basic invariants
| Dimension: | $5$ |
| Group: | $S_5$ |
| Conductor: | \(340882369\)\(\medspace = 37^{2} \cdot 499^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.3.18463.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_5$ |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.3.18463.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - x^{4} - 3x^{3} + 4x^{2} + x - 1 \)
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The roots of $f$ are computed in $\Q_{ 61 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 16 + 32\cdot 61 + 58\cdot 61^{2} + 21\cdot 61^{3} + 27\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 32 + 2\cdot 61 + 22\cdot 61^{2} + 4\cdot 61^{3} + 46\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 39 + 29\cdot 61 + 55\cdot 61^{2} + 32\cdot 61^{3} + 17\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 47 + 8\cdot 61 + 32\cdot 61^{2} + 12\cdot 61^{3} + 6\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 50 + 48\cdot 61 + 14\cdot 61^{2} + 50\cdot 61^{3} + 24\cdot 61^{4} +O(61^{5})\)
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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 |
| $1$ | $1$ | $()$ | $5$ |
| $10$ | $2$ | $(1,2)$ | $1$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $20$ | $3$ | $(1,2,3)$ | $-1$ |
| $30$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $24$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.