Basic invariants
Dimension: | $6$ |
Group: | $S_7$ |
Conductor: | \(252071\)\(\medspace = 83 \cdot 3037 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 7.1.252071.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_7$ |
Parity: | odd |
Determinant: | 1.252071.2t1.a.a |
Projective image: | $S_7$ |
Projective stem field: | Galois closure of 7.1.252071.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{7} - x^{6} + 2x^{4} - x^{3} - 2x^{2} + x + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 167 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 167 }$: \( x^{2} + 166x + 5 \)
Roots:
$r_{ 1 }$ | $=$ | \( 144 + 16\cdot 167 + 146\cdot 167^{2} + 130\cdot 167^{3} + 61\cdot 167^{4} +O(167^{5})\) |
$r_{ 2 }$ | $=$ | \( 58 + 84\cdot 167 + 31\cdot 167^{2} + 68\cdot 167^{3} + 89\cdot 167^{4} +O(167^{5})\) |
$r_{ 3 }$ | $=$ | \( 83 a + 16 + \left(72 a + 80\right)\cdot 167 + \left(71 a + 93\right)\cdot 167^{2} + \left(14 a + 104\right)\cdot 167^{3} + \left(48 a + 10\right)\cdot 167^{4} +O(167^{5})\) |
$r_{ 4 }$ | $=$ | \( 35 a + 41 + \left(126 a + 134\right)\cdot 167 + \left(62 a + 7\right)\cdot 167^{2} + \left(129 a + 10\right)\cdot 167^{3} + \left(41 a + 148\right)\cdot 167^{4} +O(167^{5})\) |
$r_{ 5 }$ | $=$ | \( 84 a + 99 + \left(94 a + 69\right)\cdot 167 + \left(95 a + 92\right)\cdot 167^{2} + \left(152 a + 47\right)\cdot 167^{3} + \left(118 a + 44\right)\cdot 167^{4} +O(167^{5})\) |
$r_{ 6 }$ | $=$ | \( 132 a + 76 + \left(40 a + 58\right)\cdot 167 + \left(104 a + 111\right)\cdot 167^{2} + \left(37 a + 76\right)\cdot 167^{3} + \left(125 a + 60\right)\cdot 167^{4} +O(167^{5})\) |
$r_{ 7 }$ | $=$ | \( 68 + 57\cdot 167 + 18\cdot 167^{2} + 63\cdot 167^{3} + 86\cdot 167^{4} +O(167^{5})\) |
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.