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:
| $r_{ 1 }$ |
$=$ |
$ 97 a + 77 + \left(40 a + 81\right)\cdot 127 + \left(86 a + 81\right)\cdot 127^{2} + \left(123 a + 68\right)\cdot 127^{3} + \left(4 a + 66\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 114 a + 53 + \left(45 a + 117\right)\cdot 127 + \left(66 a + 125\right)\cdot 127^{2} + \left(5 a + 93\right)\cdot 127^{3} + \left(20 a + 16\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 25 a + 71 + \left(27 a + 52\right)\cdot 127 + \left(120 a + 30\right)\cdot 127^{2} + \left(68 a + 65\right)\cdot 127^{3} + \left(74 a + 29\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 13 a + 40 + \left(81 a + 49\right)\cdot 127 + \left(60 a + 19\right)\cdot 127^{2} + \left(121 a + 33\right)\cdot 127^{3} + \left(106 a + 31\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 102 a + 96 + \left(99 a + 54\right)\cdot 127 + \left(6 a + 123\right)\cdot 127^{2} + \left(58 a + 13\right)\cdot 127^{3} + \left(52 a + 35\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 30 a + 47 + \left(86 a + 25\right)\cdot 127 + 40 a\cdot 127^{2} + \left(3 a + 106\right)\cdot 127^{3} + \left(122 a + 74\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5,6)$ |
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.
