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:
| $r_{ 1 }$ |
$=$ |
$ 16 a + 1 + \left(106 a + 42\right)\cdot 131 + \left(66 a + 50\right)\cdot 131^{2} + \left(104 a + 109\right)\cdot 131^{3} + \left(76 a + 127\right)\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 108 + 112\cdot 131 + 5\cdot 131^{2} + 120\cdot 131^{3} + 68\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 38 a + 87 + \left(76 a + 125\right)\cdot 131 + \left(81 a + 101\right)\cdot 131^{2} + \left(76 a + 12\right)\cdot 131^{3} + \left(119 a + 5\right)\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 25 + 55\cdot 131 + 64\cdot 131^{2} + 107\cdot 131^{3} + 108\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 93 a + 108 + \left(54 a + 130\right)\cdot 131 + \left(49 a + 89\right)\cdot 131^{2} + \left(54 a + 106\right)\cdot 131^{3} + \left(11 a + 13\right)\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 115 a + 65 + \left(24 a + 57\right)\cdot 131 + \left(64 a + 80\right)\cdot 131^{2} + \left(26 a + 67\right)\cdot 131^{3} + \left(54 a + 68\right)\cdot 131^{4} +O\left(131^{ 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 value |
| $1$ | $1$ | $()$ | $5$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $3$ |
| $15$ | $2$ | $(1,2)$ | $-1$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $2$ |
| $40$ | $3$ | $(1,2,3)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)$ | $1$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
