Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 137 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 137 }$: $ x^{2} + 131 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 101 a + 130 + \left(124 a + 82\right)\cdot 137 + \left(103 a + 101\right)\cdot 137^{2} + \left(90 a + 90\right)\cdot 137^{3} + \left(134 a + 47\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 4 + 46\cdot 137 + 109\cdot 137^{2} + 101\cdot 137^{3} + 15\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 23 a + 34 + \left(41 a + 93\right)\cdot 137 + \left(133 a + 91\right)\cdot 137^{2} + \left(4 a + 17\right)\cdot 137^{3} + \left(82 a + 36\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 114 a + 35 + \left(95 a + 43\right)\cdot 137 + \left(3 a + 28\right)\cdot 137^{2} + \left(132 a + 51\right)\cdot 137^{3} + \left(54 a + 112\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 25 a + 72 + \left(6 a + 112\right)\cdot 137 + \left(88 a + 26\right)\cdot 137^{2} + \left(123 a + 30\right)\cdot 137^{3} + \left(106 a + 6\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 36 a + 51 + \left(12 a + 45\right)\cdot 137 + \left(33 a + 52\right)\cdot 137^{2} + \left(46 a + 120\right)\cdot 137^{3} + \left(2 a + 79\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 112 a + 85 + \left(130 a + 124\right)\cdot 137 + 48 a\cdot 137^{2} + \left(13 a + 136\right)\cdot 137^{3} + \left(30 a + 112\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 7 }$
| Cycle notation |
| $(1,2,3,4,5,6,7)$ |
| $(1,2)$ |
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.
