Basic invariants
| Dimension: | $6$ |
| Group: | $S_7$ |
| Conductor: | \(1047017\)\(\medspace = 41 \cdot 25537 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.3.1047017.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_7$ |
| Parity: | even |
| Determinant: | 1.1047017.2t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.3.1047017.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - 2x^{5} - 2x^{4} + x^{3} + x^{2} + x + 1 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 9 + 122\cdot 191 + 13\cdot 191^{2} + 80\cdot 191^{3} + 123\cdot 191^{4} +O(191^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 164 + 101\cdot 191 + 54\cdot 191^{2} + 4\cdot 191^{3} + 182\cdot 191^{4} +O(191^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 151 + 11\cdot 191 + 81\cdot 191^{2} + 74\cdot 191^{3} + 118\cdot 191^{4} +O(191^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 68 a + 76 + \left(94 a + 116\right)\cdot 191 + \left(183 a + 115\right)\cdot 191^{2} + \left(163 a + 18\right)\cdot 191^{3} + \left(126 a + 72\right)\cdot 191^{4} +O(191^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 179 + 102\cdot 191 + 171\cdot 191^{2} + 138\cdot 191^{3} + 69\cdot 191^{4} +O(191^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 123 a + 144 + \left(96 a + 142\right)\cdot 191 + \left(7 a + 13\right)\cdot 191^{2} + \left(27 a + 190\right)\cdot 191^{3} + \left(64 a + 34\right)\cdot 191^{4} +O(191^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 41 + 166\cdot 191 + 122\cdot 191^{2} + 66\cdot 191^{3} + 163\cdot 191^{4} +O(191^{5})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 7 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 7 }$ | Character value |
| $1$ | $1$ | $()$ | $6$ |
| $21$ | $2$ | $(1,2)$ | $4$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $2$ |
| $70$ | $3$ | $(1,2,3)$ | $3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $210$ | $4$ | $(1,2,3,4)$ | $2$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $-1$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $-1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.