Basic invariants
| Dimension: | $10$ |
| Group: | $S_6$ |
| Conductor: | \(115\!\cdots\!729\)\(\medspace = 3^{6} \cdot 131^{4} \cdot 271^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.4.958527.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 30T164 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_6$ |
| Projective stem field: | Galois closure of 6.4.958527.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - 3x^{4} + 2x^{2} + 4x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 191 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 191 }$:
\( x^{2} + 190x + 19 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 170 a + 39 + \left(103 a + 182\right)\cdot 191 + \left(120 a + 59\right)\cdot 191^{2} + \left(169 a + 113\right)\cdot 191^{3} + \left(112 a + 13\right)\cdot 191^{4} +O(191^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 21 a + 18 + \left(87 a + 116\right)\cdot 191 + \left(70 a + 76\right)\cdot 191^{2} + \left(21 a + 162\right)\cdot 191^{3} + \left(78 a + 147\right)\cdot 191^{4} +O(191^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 5 + 9\cdot 191 + 125\cdot 191^{2} + 177\cdot 191^{3} + 92\cdot 191^{4} +O(191^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 152 a + 28 + \left(133 a + 82\right)\cdot 191 + \left(113 a + 173\right)\cdot 191^{2} + \left(153 a + 152\right)\cdot 191^{3} + \left(61 a + 155\right)\cdot 191^{4} +O(191^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 39 a + 180 + \left(57 a + 63\right)\cdot 191 + \left(77 a + 153\right)\cdot 191^{2} + \left(37 a + 1\right)\cdot 191^{3} + \left(129 a + 64\right)\cdot 191^{4} +O(191^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 113 + 119\cdot 191 + 175\cdot 191^{2} + 155\cdot 191^{3} + 98\cdot 191^{4} +O(191^{5})\)
|
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 | Complex conjugation |
| $1$ | $1$ | $()$ | $10$ | |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-2$ | |
| $15$ | $2$ | $(1,2)$ | $2$ | ✓ |
| $45$ | $2$ | $(1,2)(3,4)$ | $-2$ | |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $1$ | |
| $40$ | $3$ | $(1,2,3)$ | $1$ | |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ | |
| $90$ | $4$ | $(1,2,3,4)$ | $0$ | |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ | |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $1$ | |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |