Basic invariants
Dimension: | $35$ |
Group: | $S_7$ |
Conductor: | \(629\!\cdots\!343\)\(\medspace = 7^{18} \cdot 31^{15} \cdot 353^{15} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 7.1.536207.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 70 |
Parity: | odd |
Determinant: | 1.10943.2t1.a.a |
Projective image: | $S_7$ |
Projective stem field: | Galois closure of 7.1.536207.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{7} - x^{6} + 2x^{5} + x^{4} + 2x^{2} + x + 1 \) . |
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:
$r_{ 1 }$ | $=$ | \( 60 a + 1 + \left(105 a + 139\right)\cdot 191 + \left(176 a + 128\right)\cdot 191^{2} + \left(42 a + 3\right)\cdot 191^{3} + \left(20 a + 178\right)\cdot 191^{4} +O(191^{5})\) |
$r_{ 2 }$ | $=$ | \( 108 a + 37 + \left(23 a + 160\right)\cdot 191 + \left(40 a + 82\right)\cdot 191^{2} + \left(179 a + 169\right)\cdot 191^{3} + \left(76 a + 182\right)\cdot 191^{4} +O(191^{5})\) |
$r_{ 3 }$ | $=$ | \( 45 + 72\cdot 191 + 169\cdot 191^{2} + 35\cdot 191^{3} + 33\cdot 191^{4} +O(191^{5})\) |
$r_{ 4 }$ | $=$ | \( 83 a + 145 + \left(167 a + 75\right)\cdot 191 + \left(150 a + 99\right)\cdot 191^{2} + \left(11 a + 117\right)\cdot 191^{3} + \left(114 a + 80\right)\cdot 191^{4} +O(191^{5})\) |
$r_{ 5 }$ | $=$ | \( 10 + 50\cdot 191 + 99\cdot 191^{2} + 181\cdot 191^{3} + 41\cdot 191^{4} +O(191^{5})\) |
$r_{ 6 }$ | $=$ | \( 131 a + 61 + \left(85 a + 184\right)\cdot 191 + \left(14 a + 8\right)\cdot 191^{2} + \left(148 a + 61\right)\cdot 191^{3} + \left(170 a + 155\right)\cdot 191^{4} +O(191^{5})\) |
$r_{ 7 }$ | $=$ | \( 84 + 82\cdot 191 + 175\cdot 191^{2} + 3\cdot 191^{3} + 92\cdot 191^{4} +O(191^{5})\) |
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 |
$1$ | $1$ | $()$ | $35$ |
$21$ | $2$ | $(1,2)$ | $5$ |
$105$ | $2$ | $(1,2)(3,4)(5,6)$ | $1$ |
$105$ | $2$ | $(1,2)(3,4)$ | $-1$ |
$70$ | $3$ | $(1,2,3)$ | $-1$ |
$280$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
$210$ | $4$ | $(1,2,3,4)$ | $-1$ |
$630$ | $4$ | $(1,2,3,4)(5,6)$ | $1$ |
$504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
$210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
$420$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
$840$ | $6$ | $(1,2,3,4,5,6)$ | $1$ |
$720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
$504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $0$ |
$420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.