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:
| $r_{ 1 }$ |
$=$ |
$ 128 + 104\cdot 131 + 67\cdot 131^{2} + 47\cdot 131^{3} + 2\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 18 a + 12 + \left(101 a + 56\right)\cdot 131 + \left(67 a + 25\right)\cdot 131^{2} + \left(98 a + 127\right)\cdot 131^{3} + \left(63 a + 119\right)\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 113 a + 84 + \left(29 a + 49\right)\cdot 131 + \left(63 a + 64\right)\cdot 131^{2} + \left(32 a + 60\right)\cdot 131^{3} + \left(67 a + 14\right)\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 6 a + 81 + \left(57 a + 10\right)\cdot 131 + \left(6 a + 104\right)\cdot 131^{2} + \left(64 a + 97\right)\cdot 131^{3} + \left(12 a + 51\right)\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 116 + 69\cdot 131 + 58\cdot 131^{2} + 105\cdot 131^{3} + 35\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 125 a + 105 + \left(73 a + 101\right)\cdot 131 + \left(124 a + 72\right)\cdot 131^{2} + \left(66 a + 85\right)\cdot 131^{3} + \left(118 a + 37\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 |
| $1$ | $1$ | $()$ | $10$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-2$ |
| $15$ | $2$ | $(1,2)$ | $2$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $-2$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $1$ |
| $40$ | $3$ | $(1,2,3)$ | $1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $90$ | $4$ | $(1,2,3,4)$ | $0$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $1$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
