Basic invariants
| Dimension: | $6$ |
| Group: | $S_7$ |
| Conductor: | \(184607\) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.1.184607.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_7$ |
| 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 |
| $1$ | $1$ | $()$ | $6$ |
| $21$ | $2$ | $(1,2)$ | $4$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $2$ |
| $70$ | $3$ | $(1,2,3)$ | $3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $210$ | $4$ | $(1,2,3,4)$ | $2$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $-1$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $-1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.