Basic invariants
Dimension: | $14$ |
Group: | $S_7$ |
Conductor: | \(249\!\cdots\!007\)\(\medspace = 184607^{9} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 7.1.184607.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 30T565 |
Parity: | odd |
Determinant: | 1.184607.2t1.a.a |
Projective image: | $S_7$ |
Projective stem field: | Galois closure of 7.1.184607.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{7} - x^{6} - x^{5} + x^{4} - x^{2} + x + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 103 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 103 }$: \( x^{2} + 102x + 5 \)
Roots:
$r_{ 1 }$ | $=$ | \( 39 + 12\cdot 103 + 56\cdot 103^{2} + 59\cdot 103^{3} + 33\cdot 103^{4} +O(103^{5})\) |
$r_{ 2 }$ | $=$ | \( 86 + 6\cdot 103 + 91\cdot 103^{2} + 65\cdot 103^{3} + 35\cdot 103^{4} +O(103^{5})\) |
$r_{ 3 }$ | $=$ | \( 80 a + 78 + \left(100 a + 73\right)\cdot 103 + \left(57 a + 89\right)\cdot 103^{2} + \left(11 a + 79\right)\cdot 103^{3} + \left(19 a + 31\right)\cdot 103^{4} +O(103^{5})\) |
$r_{ 4 }$ | $=$ | \( 4 a + 14 + \left(22 a + 94\right)\cdot 103 + \left(53 a + 72\right)\cdot 103^{2} + \left(57 a + 83\right)\cdot 103^{3} + \left(97 a + 9\right)\cdot 103^{4} +O(103^{5})\) |
$r_{ 5 }$ | $=$ | \( 99 a + 18 + \left(80 a + 9\right)\cdot 103 + \left(49 a + 1\right)\cdot 103^{2} + \left(45 a + 88\right)\cdot 103^{3} + \left(5 a + 49\right)\cdot 103^{4} +O(103^{5})\) |
$r_{ 6 }$ | $=$ | \( 23 a + 55 + \left(2 a + 94\right)\cdot 103 + \left(45 a + 46\right)\cdot 103^{2} + \left(91 a + 33\right)\cdot 103^{3} + \left(83 a + 39\right)\cdot 103^{4} +O(103^{5})\) |
$r_{ 7 }$ | $=$ | \( 20 + 18\cdot 103 + 54\cdot 103^{2} + 103^{3} + 6\cdot 103^{4} +O(103^{5})\) |
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 |
$1$ | $1$ | $()$ | $14$ |
$21$ | $2$ | $(1,2)$ | $-4$ |
$105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
$105$ | $2$ | $(1,2)(3,4)$ | $2$ |
$70$ | $3$ | $(1,2,3)$ | $-1$ |
$280$ | $3$ | $(1,2,3)(4,5,6)$ | $2$ |
$210$ | $4$ | $(1,2,3,4)$ | $2$ |
$630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
$504$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
$210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
$420$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
$840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
$720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
$504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $1$ |
$420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.