Basic invariants
| Dimension: | $15$ |
| Group: | $S_7$ |
| Conductor: | \(114\!\cdots\!431\)\(\medspace = 11^{8} \cdot 3511^{5} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.1.424831.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 42T412 |
| Parity: | odd |
| Determinant: | 1.3511.2t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.1.424831.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - x^{6} - x^{4} + 2x^{3} - x^{2} + 2x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 163 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 163 }$:
\( x^{2} + 159x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 141 a + 19 + \left(98 a + 97\right)\cdot 163 + \left(38 a + 61\right)\cdot 163^{2} + 52\cdot 163^{3} + \left(67 a + 62\right)\cdot 163^{4} +O(163^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 35 a + 152 + \left(91 a + 114\right)\cdot 163 + \left(59 a + 130\right)\cdot 163^{2} + \left(52 a + 130\right)\cdot 163^{3} + \left(153 a + 93\right)\cdot 163^{4} +O(163^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 22 a + 94 + \left(64 a + 25\right)\cdot 163 + \left(124 a + 117\right)\cdot 163^{2} + \left(162 a + 14\right)\cdot 163^{3} + \left(95 a + 4\right)\cdot 163^{4} +O(163^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 128 a + 129 + \left(71 a + 118\right)\cdot 163 + \left(103 a + 114\right)\cdot 163^{2} + \left(110 a + 117\right)\cdot 163^{3} + \left(9 a + 2\right)\cdot 163^{4} +O(163^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 92 a + 70 + \left(122 a + 57\right)\cdot 163 + \left(55 a + 5\right)\cdot 163^{2} + \left(146 a + 113\right)\cdot 163^{3} + \left(83 a + 37\right)\cdot 163^{4} +O(163^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 77 + 108\cdot 163 + 116\cdot 163^{2} + 69\cdot 163^{3} + 61\cdot 163^{4} +O(163^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 71 a + 112 + \left(40 a + 129\right)\cdot 163 + \left(107 a + 105\right)\cdot 163^{2} + \left(16 a + 153\right)\cdot 163^{3} + \left(79 a + 63\right)\cdot 163^{4} +O(163^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $15$ |
| $21$ | $2$ | $(1,2)$ | $5$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $-3$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $70$ | $3$ | $(1,2,3)$ | $3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $210$ | $4$ | $(1,2,3,4)$ | $1$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $1$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $0$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.