Basic invariants
| Dimension: | $5$ |
| Group: | $S_6$ |
| Conductor: | \(39269\)\(\medspace = 107 \cdot 367 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.39269.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_6$ |
| Parity: | even |
| Determinant: | 1.39269.2t1.a.a |
| Projective image: | $S_6$ |
| Projective stem field: | Galois closure of 6.2.39269.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - 2x^{4} + 2x^{3} - 2x + 1 \)
|
The roots of $f$ are computed in $\Q_{ 577 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 135 + 47\cdot 577 + 291\cdot 577^{2} + 559\cdot 577^{3} + 163\cdot 577^{4} +O(577^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 197 + 340\cdot 577 + 441\cdot 577^{2} + 384\cdot 577^{3} + 239\cdot 577^{4} +O(577^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 202 + 5\cdot 577 + 94\cdot 577^{2} + 575\cdot 577^{3} + 150\cdot 577^{4} +O(577^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 368 + 55\cdot 577 + 510\cdot 577^{2} + 404\cdot 577^{3} +O(577^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 382 + 576\cdot 577 + 564\cdot 577^{2} + 65\cdot 577^{3} + 262\cdot 577^{4} +O(577^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 448 + 128\cdot 577 + 406\cdot 577^{2} + 317\cdot 577^{3} + 336\cdot 577^{4} +O(577^{5})\)
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Generators of the action on the roots $r_1, \ldots, r_{ 6 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 6 }$ | Character value |
| $1$ | $1$ | $()$ | $5$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-1$ |
| $15$ | $2$ | $(1,2)$ | $3$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $40$ | $3$ | $(1,2,3)$ | $2$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)$ | $1$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $-1$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.