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:
| $r_{ 1 }$ |
$=$ |
$ 109 a + 73 + \left(39 a + 123\right)\cdot 127 + \left(120 a + 51\right)\cdot 127^{2} + \left(2 a + 104\right)\cdot 127^{3} + \left(16 a + 82\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 86 a + 9 + \left(75 a + 50\right)\cdot 127 + \left(118 a + 71\right)\cdot 127^{2} + \left(36 a + 16\right)\cdot 127^{3} + \left(35 a + 34\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 41 a + 95 + \left(51 a + 39\right)\cdot 127 + \left(8 a + 114\right)\cdot 127^{2} + \left(90 a + 61\right)\cdot 127^{3} + \left(91 a + 32\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 44 a + 65 + \left(11 a + 9\right)\cdot 127 + \left(14 a + 88\right)\cdot 127^{2} + \left(35 a + 8\right)\cdot 127^{3} + \left(79 a + 117\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 83 a + 109 + \left(115 a + 103\right)\cdot 127 + \left(112 a + 90\right)\cdot 127^{2} + \left(91 a + 29\right)\cdot 127^{3} + \left(47 a + 34\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 18 a + 55 + \left(87 a + 54\right)\cdot 127 + \left(6 a + 5\right)\cdot 127^{2} + \left(124 a + 114\right)\cdot 127^{3} + \left(110 a + 95\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 102 + 126\cdot 127 + 85\cdot 127^{2} + 45\cdot 127^{3} + 111\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 7 }$
| Cycle notation |
| $(1,2,3,4,5,6,7)$ |
| $(1,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 7 }$
| Character value |
| $1$ | $1$ | $()$ | $14$ |
| $21$ | $2$ | $(1,2)$ | $-6$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $-2$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $2$ |
| $70$ | $3$ | $(1,2,3)$ | $2$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $210$ | $4$ | $(1,2,3,4)$ | $0$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $2$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $1$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $-1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
