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:
| $r_{ 1 }$ |
$=$ |
$ 18 + 133\cdot 137 + 23\cdot 137^{2} + 120\cdot 137^{3} + 75\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 88 a + 19 + \left(119 a + 49\right)\cdot 137 + \left(95 a + 82\right)\cdot 137^{2} + \left(93 a + 123\right)\cdot 137^{3} + \left(82 a + 36\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 49 a + 136 + \left(17 a + 130\right)\cdot 137 + \left(41 a + 126\right)\cdot 137^{2} + \left(43 a + 41\right)\cdot 137^{3} + \left(54 a + 28\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 126 a + 6 + \left(61 a + 124\right)\cdot 137 + \left(2 a + 59\right)\cdot 137^{2} + \left(69 a + 105\right)\cdot 137^{3} + \left(105 a + 7\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 11 a + 77 + \left(75 a + 95\right)\cdot 137 + \left(134 a + 12\right)\cdot 137^{2} + \left(67 a + 106\right)\cdot 137^{3} + \left(31 a + 23\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 20 + 15\cdot 137 + 105\cdot 137^{2} + 50\cdot 137^{3} + 101\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,3)$ |
| $(1,2)(3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$9$ |
| $45$ |
$2$ |
$(1,2)(3,4)$ |
$1$ |
| $40$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$0$ |
| $40$ |
$3$ |
$(1,2,3)$ |
$0$ |
| $90$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$1$ |
| $72$ |
$5$ |
$(1,2,3,4,5)$ |
$-1$ |
| $72$ |
$5$ |
$(1,3,4,5,2)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
