Basic invariants
Dimension: | $6$ |
Group: | $S_7$ |
Conductor: | \(594751\) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 7.1.594751.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_7$ |
Parity: | odd |
Determinant: | 1.594751.2t1.a.a |
Projective image: | $S_7$ |
Projective stem field: | Galois closure of 7.1.594751.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{7} - x^{6} - x^{4} + 2x^{3} - x^{2} - 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 127 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 127 }$: \( x^{2} + 126x + 3 \)
Roots:
$r_{ 1 }$ | $=$ | \( 84 + 93\cdot 127 + 126\cdot 127^{2} + 92\cdot 127^{3} + 36\cdot 127^{4} +O(127^{5})\) |
$r_{ 2 }$ | $=$ | \( 38 + 35\cdot 127 + 40\cdot 127^{2} + 63\cdot 127^{3} + 116\cdot 127^{4} +O(127^{5})\) |
$r_{ 3 }$ | $=$ | \( 14 a + 91 + \left(31 a + 45\right)\cdot 127 + \left(27 a + 88\right)\cdot 127^{2} + \left(87 a + 99\right)\cdot 127^{3} + \left(68 a + 25\right)\cdot 127^{4} +O(127^{5})\) |
$r_{ 4 }$ | $=$ | \( 68 + 85\cdot 127 + 51\cdot 127^{2} + 41\cdot 127^{3} + 3\cdot 127^{4} +O(127^{5})\) |
$r_{ 5 }$ | $=$ | \( 113 a + 105 + \left(95 a + 62\right)\cdot 127 + \left(99 a + 84\right)\cdot 127^{2} + \left(39 a + 32\right)\cdot 127^{3} + \left(58 a + 7\right)\cdot 127^{4} +O(127^{5})\) |
$r_{ 6 }$ | $=$ | \( 53 a + 35 + \left(50 a + 30\right)\cdot 127 + \left(77 a + 108\right)\cdot 127^{2} + \left(105 a + 74\right)\cdot 127^{3} + \left(52 a + 58\right)\cdot 127^{4} +O(127^{5})\) |
$r_{ 7 }$ | $=$ | \( 74 a + 88 + \left(76 a + 27\right)\cdot 127 + \left(49 a + 8\right)\cdot 127^{2} + \left(21 a + 103\right)\cdot 127^{3} + \left(74 a + 5\right)\cdot 127^{4} +O(127^{5})\) |
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.