Basic invariants
| Dimension: | $6$ |
| Group: | $S_7$ |
| Conductor: | \(286711\) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 7.1.286711.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_7$ |
| Parity: | odd |
| Projective image: | $S_7$ |
| Projective field: | Galois closure of 7.1.286711.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 191 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 191 }$:
\( x^{2} + 190x + 19 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 98 a + 183 + \left(61 a + 66\right)\cdot 191 + \left(127 a + 123\right)\cdot 191^{2} + \left(49 a + 32\right)\cdot 191^{3} + \left(44 a + 122\right)\cdot 191^{4} +O(191^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 183 a + \left(25 a + 87\right)\cdot 191 + \left(105 a + 67\right)\cdot 191^{2} + \left(20 a + 165\right)\cdot 191^{3} + \left(101 a + 38\right)\cdot 191^{4} +O(191^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 8 a + 183 + \left(165 a + 120\right)\cdot 191 + \left(85 a + 146\right)\cdot 191^{2} + \left(170 a + 80\right)\cdot 191^{3} + \left(89 a + 119\right)\cdot 191^{4} +O(191^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 188 a + 131 + \left(108 a + 105\right)\cdot 191 + \left(181 a + 2\right)\cdot 191^{2} + \left(72 a + 37\right)\cdot 191^{3} + \left(152 a + 115\right)\cdot 191^{4} +O(191^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 3 a + 128 + \left(82 a + 26\right)\cdot 191 + \left(9 a + 75\right)\cdot 191^{2} + \left(118 a + 119\right)\cdot 191^{3} + \left(38 a + 3\right)\cdot 191^{4} +O(191^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 49 + 135\cdot 191 + 159\cdot 191^{2} + 182\cdot 191^{3} + 56\cdot 191^{4} +O(191^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 93 a + 90 + \left(129 a + 30\right)\cdot 191 + \left(63 a + 189\right)\cdot 191^{2} + \left(141 a + 145\right)\cdot 191^{3} + \left(146 a + 116\right)\cdot 191^{4} +O(191^{5})\)
|
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 values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $6$ |
| $21$ | $2$ | $(1,2)$ | $4$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $2$ |
| $70$ | $3$ | $(1,2,3)$ | $3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $210$ | $4$ | $(1,2,3,4)$ | $2$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $-1$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $-1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |