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:
| $r_{ 1 }$ |
$=$ |
$ 118 a + 5 + \left(93 a + 77\right)\cdot 137 + \left(97 a + 36\right)\cdot 137^{2} + \left(50 a + 102\right)\cdot 137^{3} + \left(121 a + 19\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 19 a + 28 + \left(43 a + 111\right)\cdot 137 + \left(39 a + 117\right)\cdot 137^{2} + \left(86 a + 34\right)\cdot 137^{3} + \left(15 a + 12\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 51 + 99\cdot 137 + 51\cdot 137^{2} + 46\cdot 137^{3} + 97\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 29 a + 88 + \left(21 a + 131\right)\cdot 137 + \left(51 a + 24\right)\cdot 137^{2} + \left(96 a + 68\right)\cdot 137^{3} + \left(91 a + 92\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 116 + 35\cdot 137 + 6\cdot 137^{2} + 112\cdot 137^{3} + 53\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 108 a + 125 + \left(115 a + 92\right)\cdot 137 + \left(85 a + 36\right)\cdot 137^{2} + \left(40 a + 47\right)\cdot 137^{3} + \left(45 a + 135\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$5$ |
| $15$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$-1$ |
| $15$ |
$2$ |
$(1,2)$ |
$3$ |
| $45$ |
$2$ |
$(1,2)(3,4)$ |
$1$ |
| $40$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$-1$ |
| $40$ |
$3$ |
$(1,2,3)$ |
$2$ |
| $90$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$-1$ |
| $90$ |
$4$ |
$(1,2,3,4)$ |
$1$ |
| $144$ |
$5$ |
$(1,2,3,4,5)$ |
$0$ |
| $120$ |
$6$ |
$(1,2,3,4,5,6)$ |
$-1$ |
| $120$ |
$6$ |
$(1,2,3)(4,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
