Basic invariants
| Dimension: | $6$ |
| Group: | $S_5$ |
| Conductor: | \(1630033565404672\)\(\medspace = 2^{9} \cdot 47^{3} \cdot 313^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.5.117688.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 20T30 |
| Parity: | even |
| Determinant: | 1.117688.2t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.5.117688.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - x^{4} - 5x^{3} + 4x^{2} + 4x - 1 \)
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The roots of $f$ are computed in $\Q_{ 401 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 150 + 146\cdot 401 + 320\cdot 401^{2} + 69\cdot 401^{3} + 339\cdot 401^{4} +O(401^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 191 + 4\cdot 401 + 236\cdot 401^{2} + 140\cdot 401^{3} + 244\cdot 401^{4} +O(401^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 242 + 126\cdot 401 + 319\cdot 401^{2} + 324\cdot 401^{3} + 153\cdot 401^{4} +O(401^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 296 + 372\cdot 401 + 118\cdot 401^{2} + 274\cdot 401^{3} + 66\cdot 401^{4} +O(401^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 325 + 151\cdot 401 + 208\cdot 401^{2} + 393\cdot 401^{3} + 398\cdot 401^{4} +O(401^{5})\)
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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 |
| $1$ | $1$ | $()$ | $6$ |
| $10$ | $2$ | $(1,2)$ | $0$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $-2$ |
| $20$ | $3$ | $(1,2,3)$ | $0$ |
| $30$ | $4$ | $(1,2,3,4)$ | $0$ |
| $24$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.