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:
| $r_{ 1 }$ |
$=$ |
$ 33 a + 67 + \left(92 a + 35\right)\cdot 137 + \left(101 a + 9\right)\cdot 137^{2} + \left(16 a + 42\right)\cdot 137^{3} + \left(84 a + 132\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 10 a + 9 + \left(37 a + 97\right)\cdot 137 + \left(34 a + 50\right)\cdot 137^{2} + \left(11 a + 30\right)\cdot 137^{3} + \left(128 a + 128\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 127 a + 69 + \left(99 a + 35\right)\cdot 137 + \left(102 a + 82\right)\cdot 137^{2} + \left(125 a + 63\right)\cdot 137^{3} + \left(8 a + 63\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 43 + 8\cdot 137 + 92\cdot 137^{2} + 70\cdot 137^{3} + 74\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 104 a + 128 + \left(44 a + 7\right)\cdot 137 + \left(35 a + 116\right)\cdot 137^{2} + \left(120 a + 40\right)\cdot 137^{3} + \left(52 a + 72\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 95 + 89\cdot 137 + 60\cdot 137^{2} + 26\cdot 137^{3} + 77\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 value |
| $1$ | $1$ | $()$ | $16$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $15$ | $2$ | $(1,2)$ | $0$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-2$ |
| $40$ | $3$ | $(1,2,3)$ | $-2$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $90$ | $4$ | $(1,2,3,4)$ | $0$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
