Basic invariants
Dimension: | $6$ |
Group: | $S_7$ |
Conductor: | \(286711\) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 7.1.286711.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_7$ |
Parity: | odd |
Determinant: | 1.286711.2t1.a.a |
Projective image: | $S_7$ |
Projective stem field: | Galois closure of 7.1.286711.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{7} - x^{5} - 2x^{4} + x^{3} + 2x^{2} - x - 1 \) . |
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:
$r_{ 1 }$ | $=$ | \( 98 a + 183 + \left(61 a + 66\right)\cdot 191 + \left(127 a + 123\right)\cdot 191^{2} + \left(49 a + 32\right)\cdot 191^{3} + \left(44 a + 122\right)\cdot 191^{4} +O(191^{5})\) |
$r_{ 2 }$ | $=$ | \( 183 a + \left(25 a + 87\right)\cdot 191 + \left(105 a + 67\right)\cdot 191^{2} + \left(20 a + 165\right)\cdot 191^{3} + \left(101 a + 38\right)\cdot 191^{4} +O(191^{5})\) |
$r_{ 3 }$ | $=$ | \( 8 a + 183 + \left(165 a + 120\right)\cdot 191 + \left(85 a + 146\right)\cdot 191^{2} + \left(170 a + 80\right)\cdot 191^{3} + \left(89 a + 119\right)\cdot 191^{4} +O(191^{5})\) |
$r_{ 4 }$ | $=$ | \( 188 a + 131 + \left(108 a + 105\right)\cdot 191 + \left(181 a + 2\right)\cdot 191^{2} + \left(72 a + 37\right)\cdot 191^{3} + \left(152 a + 115\right)\cdot 191^{4} +O(191^{5})\) |
$r_{ 5 }$ | $=$ | \( 3 a + 128 + \left(82 a + 26\right)\cdot 191 + \left(9 a + 75\right)\cdot 191^{2} + \left(118 a + 119\right)\cdot 191^{3} + \left(38 a + 3\right)\cdot 191^{4} +O(191^{5})\) |
$r_{ 6 }$ | $=$ | \( 49 + 135\cdot 191 + 159\cdot 191^{2} + 182\cdot 191^{3} + 56\cdot 191^{4} +O(191^{5})\) |
$r_{ 7 }$ | $=$ | \( 93 a + 90 + \left(129 a + 30\right)\cdot 191 + \left(63 a + 189\right)\cdot 191^{2} + \left(141 a + 145\right)\cdot 191^{3} + \left(146 a + 116\right)\cdot 191^{4} +O(191^{5})\) |
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 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.