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:
| $r_{ 1 }$ |
$=$ |
$ 107 + 128\cdot 137 + 60\cdot 137^{2} + 58\cdot 137^{3} +O\left(137^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 126 a + 18 + \left(29 a + 91\right)\cdot 137 + \left(71 a + 89\right)\cdot 137^{2} + \left(88 a + 46\right)\cdot 137^{3} + \left(42 a + 28\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 19 + 31\cdot 137 + 13\cdot 137^{2} + 41\cdot 137^{3} + 57\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 64 + 85\cdot 137 + 119\cdot 137^{2} + 63\cdot 137^{3} + 14\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 29 + 69\cdot 137 + 99\cdot 137^{2} + 81\cdot 137^{3} + 93\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 86 + 134\cdot 137 + 88\cdot 137^{2} + 23\cdot 137^{3} + 21\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 11 a + 89 + \left(107 a + 7\right)\cdot 137 + \left(65 a + 76\right)\cdot 137^{2} + \left(48 a + 95\right)\cdot 137^{3} + \left(94 a + 58\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 |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$6$ |
| $21$ |
$2$ |
$(1,2)$ |
$4$ |
| $105$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$0$ |
| $105$ |
$2$ |
$(1,2)(3,4)$ |
$2$ |
| $70$ |
$3$ |
$(1,2,3)$ |
$3$ |
| $280$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$0$ |
| $210$ |
$4$ |
$(1,2,3,4)$ |
$2$ |
| $630$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$0$ |
| $504$ |
$5$ |
$(1,2,3,4,5)$ |
$1$ |
| $210$ |
$6$ |
$(1,2,3)(4,5)(6,7)$ |
$-1$ |
| $420$ |
$6$ |
$(1,2,3)(4,5)$ |
$1$ |
| $840$ |
$6$ |
$(1,2,3,4,5,6)$ |
$0$ |
| $720$ |
$7$ |
$(1,2,3,4,5,6,7)$ |
$-1$ |
| $504$ |
$10$ |
$(1,2,3,4,5)(6,7)$ |
$-1$ |
| $420$ |
$12$ |
$(1,2,3,4)(5,6,7)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
