Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 131 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 131 }$: $ x^{2} + 127 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 130 a + 70 + \left(52 a + 110\right)\cdot 131 + \left(83 a + 70\right)\cdot 131^{2} + \left(30 a + 107\right)\cdot 131^{3} + \left(92 a + 23\right)\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ a + 66 + \left(78 a + 61\right)\cdot 131 + \left(47 a + 89\right)\cdot 131^{2} + \left(100 a + 15\right)\cdot 131^{3} + \left(38 a + 100\right)\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 116 a + 7 + \left(114 a + 67\right)\cdot 131 + \left(110 a + 105\right)\cdot 131^{2} + \left(29 a + 36\right)\cdot 131^{3} + \left(6 a + 21\right)\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 15 a + 78 + \left(16 a + 17\right)\cdot 131 + \left(20 a + 41\right)\cdot 131^{2} + \left(101 a + 45\right)\cdot 131^{3} + \left(124 a + 16\right)\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 113 + 35\cdot 131 + 66\cdot 131^{2} + 86\cdot 131^{3} + 72\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 109 + 76\cdot 131^{2} + 74\cdot 131^{3} + 48\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 82 + 99\cdot 131 + 74\cdot 131^{2} + 26\cdot 131^{3} + 110\cdot 131^{4} +O\left(131^{ 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.
