Basic invariants
Dimension: | $6$ |
Group: | $S_7$ |
Conductor: | \(199559\) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 7.1.199559.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_7$ |
Parity: | odd |
Determinant: | 1.199559.2t1.a.a |
Projective image: | $S_7$ |
Projective stem field: | Galois closure of 7.1.199559.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{7} - x^{6} + x^{3} - x + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 193 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 193 }$: \( x^{2} + 192x + 5 \)
Roots:
$r_{ 1 }$ | $=$ | \( 23 a + 18 + \left(190 a + 60\right)\cdot 193 + \left(181 a + 46\right)\cdot 193^{2} + \left(105 a + 42\right)\cdot 193^{3} + \left(13 a + 160\right)\cdot 193^{4} +O(193^{5})\) |
$r_{ 2 }$ | $=$ | \( 57 a + 120 + \left(112 a + 18\right)\cdot 193 + \left(104 a + 55\right)\cdot 193^{2} + \left(125 a + 189\right)\cdot 193^{3} + \left(90 a + 151\right)\cdot 193^{4} +O(193^{5})\) |
$r_{ 3 }$ | $=$ | \( 177 + 7\cdot 193 + 84\cdot 193^{2} + 133\cdot 193^{3} + 169\cdot 193^{4} +O(193^{5})\) |
$r_{ 4 }$ | $=$ | \( 169 + 183\cdot 193 + 61\cdot 193^{2} + 130\cdot 193^{3} +O(193^{5})\) |
$r_{ 5 }$ | $=$ | \( 170 a + 41 + \left(2 a + 34\right)\cdot 193 + \left(11 a + 38\right)\cdot 193^{2} + \left(87 a + 159\right)\cdot 193^{3} + \left(179 a + 67\right)\cdot 193^{4} +O(193^{5})\) |
$r_{ 6 }$ | $=$ | \( 71 + 7\cdot 193 + 53\cdot 193^{2} + 100\cdot 193^{3} + 104\cdot 193^{4} +O(193^{5})\) |
$r_{ 7 }$ | $=$ | \( 136 a + 177 + \left(80 a + 73\right)\cdot 193 + \left(88 a + 47\right)\cdot 193^{2} + \left(67 a + 17\right)\cdot 193^{3} + \left(102 a + 117\right)\cdot 193^{4} +O(193^{5})\) |
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.