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:
| $r_{ 1 }$ |
$=$ |
$ 93 a + 63 + \left(81 a + 73\right)\cdot 137 + \left(78 a + 123\right)\cdot 137^{2} + \left(100 a + 64\right)\cdot 137^{3} + \left(131 a + 94\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 124 a + 136 + \left(85 a + 3\right)\cdot 137 + \left(35 a + 90\right)\cdot 137^{2} + \left(42 a + 1\right)\cdot 137^{3} + \left(107 a + 126\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 56 + 27\cdot 137 + 21\cdot 137^{2} + 129\cdot 137^{3} + 42\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 95 a + 2 + \left(8 a + 84\right)\cdot 137 + \left(100 a + 111\right)\cdot 137^{2} + \left(34 a + 59\right)\cdot 137^{3} + \left(37 a + 45\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 44 a + 73 + \left(55 a + 59\right)\cdot 137 + \left(58 a + 102\right)\cdot 137^{2} + \left(36 a + 41\right)\cdot 137^{3} + \left(5 a + 99\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 42 a + 24 + \left(128 a + 41\right)\cdot 137 + \left(36 a + 18\right)\cdot 137^{2} + \left(102 a + 31\right)\cdot 137^{3} + \left(99 a + 97\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 13 a + 58 + \left(51 a + 121\right)\cdot 137 + \left(101 a + 80\right)\cdot 137^{2} + \left(94 a + 82\right)\cdot 137^{3} + \left(29 a + 42\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 values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$14$ |
| $21$ |
$2$ |
$(1,2)$ |
$6$ |
| $105$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$2$ |
| $105$ |
$2$ |
$(1,2)(3,4)$ |
$2$ |
| $70$ |
$3$ |
$(1,2,3)$ |
$2$ |
| $280$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$-1$ |
| $210$ |
$4$ |
$(1,2,3,4)$ |
$0$ |
| $630$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$0$ |
| $504$ |
$5$ |
$(1,2,3,4,5)$ |
$-1$ |
| $210$ |
$6$ |
$(1,2,3)(4,5)(6,7)$ |
$2$ |
| $420$ |
$6$ |
$(1,2,3)(4,5)$ |
$0$ |
| $840$ |
$6$ |
$(1,2,3,4,5,6)$ |
$-1$ |
| $720$ |
$7$ |
$(1,2,3,4,5,6,7)$ |
$0$ |
| $504$ |
$10$ |
$(1,2,3,4,5)(6,7)$ |
$1$ |
| $420$ |
$12$ |
$(1,2,3,4)(5,6,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
