Basic invariants
| Dimension: | $21$ |
| Group: | $S_7$ |
| Conductor: | \(147\!\cdots\!801\)\(\medspace = 7^{10} \cdot 29633^{10} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.3.1452017.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 84 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.3.1452017.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - 2x^{5} - 2x^{4} + 4x^{2} + x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 149 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 149 }$:
\( x^{2} + 145x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 7 a + 77 + \left(56 a + 6\right)\cdot 149 + \left(66 a + 26\right)\cdot 149^{2} + \left(73 a + 28\right)\cdot 149^{3} + \left(119 a + 110\right)\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 96 a + 10 + \left(135 a + 17\right)\cdot 149 + \left(14 a + 84\right)\cdot 149^{2} + \left(109 a + 129\right)\cdot 149^{3} + \left(84 a + 142\right)\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 104 + 35\cdot 149 + 116\cdot 149^{2} + 61\cdot 149^{3} + 149^{4} +O(149^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 53 a + 96 + \left(13 a + 16\right)\cdot 149 + \left(134 a + 8\right)\cdot 149^{2} + \left(39 a + 104\right)\cdot 149^{3} + \left(64 a + 74\right)\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 23 a + 56 + \left(89 a + 130\right)\cdot 149 + \left(59 a + 62\right)\cdot 149^{2} + \left(113 a + 109\right)\cdot 149^{3} + \left(54 a + 46\right)\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 126 a + 148 + \left(59 a + 16\right)\cdot 149 + \left(89 a + 63\right)\cdot 149^{2} + \left(35 a + 56\right)\cdot 149^{3} + \left(94 a + 3\right)\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 142 a + 105 + \left(92 a + 74\right)\cdot 149 + \left(82 a + 86\right)\cdot 149^{2} + \left(75 a + 106\right)\cdot 149^{3} + \left(29 a + 67\right)\cdot 149^{4} +O(149^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $21$ |
| $21$ | $2$ | $(1,2)$ | $1$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $-3$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $70$ | $3$ | $(1,2,3)$ | $-3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $210$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.