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:
| $r_{ 1 }$ |
$=$ |
$ 125 + 104\cdot 127 + 3\cdot 127^{2} + 87\cdot 127^{3} + 73\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 22 a + 60 + \left(62 a + 105\right)\cdot 127 + \left(11 a + 113\right)\cdot 127^{2} + \left(4 a + 83\right)\cdot 127^{3} + \left(109 a + 117\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 4 a + 34 + \left(107 a + 105\right)\cdot 127 + \left(67 a + 3\right)\cdot 127^{2} + \left(76 a + 124\right)\cdot 127^{3} + \left(123 a + 42\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 31 a + 69 + \left(12 a + 55\right)\cdot 127 + \left(26 a + 45\right)\cdot 127^{2} + \left(39 a + 122\right)\cdot 127^{3} + \left(36 a + 108\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 96 a + 100 + \left(114 a + 36\right)\cdot 127 + \left(100 a + 59\right)\cdot 127^{2} + \left(87 a + 8\right)\cdot 127^{3} + \left(90 a + 106\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 105 a + 82 + \left(64 a + 18\right)\cdot 127 + \left(115 a + 63\right)\cdot 127^{2} + \left(122 a + 76\right)\cdot 127^{3} + \left(17 a + 95\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 123 a + 38 + \left(19 a + 81\right)\cdot 127 + \left(59 a + 91\right)\cdot 127^{2} + \left(50 a + 5\right)\cdot 127^{3} + \left(3 a + 90\right)\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 values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$21$ |
| $21$ |
$2$ |
$(1,2)$ |
$-1$ |
| $105$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$3$ |
| $105$ |
$2$ |
$(1,2)(3,4)$ |
$1$ |
| $70$ |
$3$ |
$(1,2,3)$ |
$-3$ |
| $280$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$0$ |
| $210$ |
$4$ |
$(1,2,3,4)$ |
$1$ |
| $630$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$-1$ |
| $504$ |
$5$ |
$(1,2,3,4,5)$ |
$1$ |
| $210$ |
$6$ |
$(1,2,3)(4,5)(6,7)$ |
$1$ |
| $420$ |
$6$ |
$(1,2,3)(4,5)$ |
$-1$ |
| $840$ |
$6$ |
$(1,2,3,4,5,6)$ |
$0$ |
| $720$ |
$7$ |
$(1,2,3,4,5,6,7)$ |
$0$ |
| $504$ |
$10$ |
$(1,2,3,4,5)(6,7)$ |
$-1$ |
| $420$ |
$12$ |
$(1,2,3,4)(5,6,7)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
