Basic invariants
| Dimension: | $6$ |
| Group: | $S_7$ |
| Conductor: | \(136\!\cdots\!507\)\(\medspace = 101^{5} \cdot 2647^{5} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.1.267347.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 14T46 |
| Parity: | odd |
| Determinant: | 1.267347.2t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.1.267347.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - x^{4} - 2x^{3} + x^{2} + x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 179 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 179 }$:
\( x^{2} + 172x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 50 a + 108 + \left(151 a + 150\right)\cdot 179 + \left(84 a + 126\right)\cdot 179^{2} + \left(46 a + 60\right)\cdot 179^{3} + \left(118 a + 165\right)\cdot 179^{4} +O(179^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 157 a + 29 + \left(160 a + 35\right)\cdot 179 + \left(20 a + 160\right)\cdot 179^{2} + \left(97 a + 167\right)\cdot 179^{3} + \left(166 a + 160\right)\cdot 179^{4} +O(179^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 157 a + 144 + \left(44 a + 35\right)\cdot 179 + \left(138 a + 126\right)\cdot 179^{2} + \left(131 a + 107\right)\cdot 179^{3} + \left(82 a + 170\right)\cdot 179^{4} +O(179^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 22 a + 54 + \left(18 a + 109\right)\cdot 179 + \left(158 a + 145\right)\cdot 179^{2} + \left(81 a + 110\right)\cdot 179^{3} + \left(12 a + 155\right)\cdot 179^{4} +O(179^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 129 a + 100 + \left(27 a + 85\right)\cdot 179 + \left(94 a + 32\right)\cdot 179^{2} + \left(132 a + 122\right)\cdot 179^{3} + \left(60 a + 51\right)\cdot 179^{4} +O(179^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 22 a + 169 + \left(134 a + 13\right)\cdot 179 + \left(40 a + 154\right)\cdot 179^{2} + \left(47 a + 175\right)\cdot 179^{3} + \left(96 a + 80\right)\cdot 179^{4} +O(179^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 112 + 106\cdot 179 + 149\cdot 179^{2} + 149\cdot 179^{3} + 109\cdot 179^{4} +O(179^{5})\)
|
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.