Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 127 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 127 }$: $ x^{2} + 126 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 12 a + 91 + \left(118 a + 55\right)\cdot 127 + \left(37 a + 122\right)\cdot 127^{2} + \left(32 a + 89\right)\cdot 127^{3} + \left(111 a + 98\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 115 a + 103 + \left(8 a + 34\right)\cdot 127 + \left(89 a + 42\right)\cdot 127^{2} + \left(94 a + 84\right)\cdot 127^{3} + \left(15 a + 50\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 46 a + 53 + \left(69 a + 18\right)\cdot 127 + \left(36 a + 126\right)\cdot 127^{2} + \left(21 a + 40\right)\cdot 127^{3} + \left(5 a + 56\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 14 + 33\cdot 127 + 105\cdot 127^{2} + 45\cdot 127^{3} + 108\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 81 a + 99 + \left(57 a + 41\right)\cdot 127 + \left(90 a + 93\right)\cdot 127^{2} + \left(105 a + 25\right)\cdot 127^{3} + \left(121 a + 40\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 23 + 70\cdot 127 + 18\cdot 127^{2} + 94\cdot 127^{3} + 26\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)$ |
| $(1,3,4)(2,5,6)$ |
| $(4,6)$ |
| $(3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,2)(3,5)(4,6)$ | $-3$ |
| $3$ | $2$ | $(1,2)$ | $1$ |
| $3$ | $2$ | $(1,2)(3,5)$ | $-1$ |
| $4$ | $3$ | $(1,3,4)(2,5,6)$ | $0$ |
| $4$ | $3$ | $(1,4,3)(2,6,5)$ | $0$ |
| $4$ | $6$ | $(1,5,6,2,3,4)$ | $0$ |
| $4$ | $6$ | $(1,4,3,2,6,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
