Basic invariants
| Dimension: | $5$ |
| Group: | $S_6$ |
| Conductor: | \(174172\)\(\medspace = 2^{2} \cdot 43543 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.174172.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_6$ |
| Parity: | even |
| Determinant: | 1.174172.2t1.a.a |
| Projective image: | $S_6$ |
| Projective stem field: | Galois closure of 6.2.174172.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - 2x^{4} + 4x^{3} - 3x - 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 331 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 331 }$:
\( x^{2} + 326x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 72 + 121\cdot 331 + 191\cdot 331^{2} + 144\cdot 331^{3} + 155\cdot 331^{4} +O(331^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 286 + 308\cdot 331 + 53\cdot 331^{2} + 59\cdot 331^{3} + 25\cdot 331^{4} +O(331^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 30 a + 40 + \left(3 a + 114\right)\cdot 331 + \left(161 a + 104\right)\cdot 331^{2} + \left(57 a + 165\right)\cdot 331^{3} + \left(145 a + 107\right)\cdot 331^{4} +O(331^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 288 + 76\cdot 331 + 63\cdot 331^{2} + 318\cdot 331^{3} + 190\cdot 331^{4} +O(331^{5})\)
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| $r_{ 5 }$ | $=$ |
\( 118 + 272\cdot 331 + 4\cdot 331^{2} + 14\cdot 331^{3} + 69\cdot 331^{4} +O(331^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 301 a + 190 + \left(327 a + 99\right)\cdot 331 + \left(169 a + 244\right)\cdot 331^{2} + \left(273 a + 291\right)\cdot 331^{3} + \left(185 a + 113\right)\cdot 331^{4} +O(331^{5})\)
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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.