Basic invariants
| Dimension: | $6$ |
| Group: | $S_7$ |
| Conductor: | \(820\!\cdots\!151\)\(\medspace = 382631^{5} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.1.382631.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 14T46 |
| Parity: | odd |
| Determinant: | 1.382631.2t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.1.382631.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - x^{5} + x^{3} - x - 1 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 2 a + 70 + \left(35 a + 89\right)\cdot 103 + \left(31 a + 99\right)\cdot 103^{2} + \left(24 a + 3\right)\cdot 103^{3} + \left(88 a + 70\right)\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 55 a + 3 + \left(50 a + 68\right)\cdot 103 + \left(39 a + 54\right)\cdot 103^{2} + \left(27 a + 21\right)\cdot 103^{3} + \left(21 a + 78\right)\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 48 a + 58 + \left(52 a + 63\right)\cdot 103 + \left(63 a + 43\right)\cdot 103^{2} + \left(75 a + 9\right)\cdot 103^{3} + \left(81 a + 72\right)\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 101 a + 72 + \left(67 a + 19\right)\cdot 103 + \left(71 a + 96\right)\cdot 103^{2} + \left(78 a + 99\right)\cdot 103^{3} + \left(14 a + 30\right)\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 19 + 14\cdot 103 + 53\cdot 103^{2} + 69\cdot 103^{3} + 50\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 101 + 77\cdot 103 + 66\cdot 103^{2} + 92\cdot 103^{3} + 70\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 89 + 78\cdot 103 + 100\cdot 103^{2} + 11\cdot 103^{3} + 39\cdot 103^{4} +O(103^{5})\)
|
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.