Basic invariants
| Dimension: | $6$ |
| Group: | $S_7$ |
| Conductor: | \(371131\) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.1.371131.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_7$ |
| Parity: | odd |
| Determinant: | 1.371131.2t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.1.371131.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - x^{4} - 2x^{2} - x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 223 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 223 }$:
\( x^{2} + 221x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 10 a + 57 + \left(101 a + 4\right)\cdot 223 + 98\cdot 223^{2} + \left(91 a + 34\right)\cdot 223^{3} + \left(125 a + 89\right)\cdot 223^{4} +O(223^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 174 a + 59 + \left(86 a + 158\right)\cdot 223 + \left(116 a + 140\right)\cdot 223^{2} + \left(25 a + 36\right)\cdot 223^{3} + \left(33 a + 162\right)\cdot 223^{4} +O(223^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 83 + 56\cdot 223 + 204\cdot 223^{2} + 218\cdot 223^{3} + 116\cdot 223^{4} +O(223^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 49 a + 184 + \left(136 a + 157\right)\cdot 223 + \left(106 a + 63\right)\cdot 223^{2} + \left(197 a + 194\right)\cdot 223^{3} + \left(189 a + 202\right)\cdot 223^{4} +O(223^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 213 a + 77 + \left(121 a + 196\right)\cdot 223 + \left(222 a + 220\right)\cdot 223^{2} + \left(131 a + 215\right)\cdot 223^{3} + \left(97 a + 25\right)\cdot 223^{4} +O(223^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 155 + 129\cdot 223 + 145\cdot 223^{2} + 45\cdot 223^{3} + 136\cdot 223^{4} +O(223^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 54 + 189\cdot 223 + 18\cdot 223^{2} + 146\cdot 223^{3} + 158\cdot 223^{4} +O(223^{5})\)
|
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.