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:
| $r_{ 1 }$ |
$=$ |
$ 9 a + 23 + \left(81 a + 79\right)\cdot 127 + \left(124 a + 82\right)\cdot 127^{2} + \left(36 a + 40\right)\cdot 127^{3} + \left(3 a + 57\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 84 a + 56 + \left(5 a + 15\right)\cdot 127 + \left(4 a + 10\right)\cdot 127^{2} + \left(62 a + 24\right)\cdot 127^{3} + \left(45 a + 69\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 118 a + 32 + \left(45 a + 24\right)\cdot 127 + \left(2 a + 126\right)\cdot 127^{2} + \left(90 a + 79\right)\cdot 127^{3} + \left(123 a + 23\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 34 a + 87 + \left(102 a + 3\right)\cdot 127 + \left(15 a + 2\right)\cdot 127^{2} + \left(109 a + 101\right)\cdot 127^{3} + \left(56 a + 45\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 43 a + 13 + \left(121 a + 64\right)\cdot 127 + \left(122 a + 8\right)\cdot 127^{2} + \left(64 a + 82\right)\cdot 127^{3} + \left(81 a + 52\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 93 a + 121 + \left(24 a + 71\right)\cdot 127 + \left(111 a + 42\right)\cdot 127^{2} + \left(17 a + 67\right)\cdot 127^{3} + \left(70 a + 120\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 50 + 122\cdot 127 + 108\cdot 127^{2} + 112\cdot 127^{3} + 11\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 |
| $1$ | $1$ | $()$ | $35$ |
| $21$ | $2$ | $(1,2)$ | $5$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $1$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $70$ | $3$ | $(1,2,3)$ | $-1$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $210$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $1$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $1$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $0$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
