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:
| $r_{ 1 }$ |
$=$ |
$ 103 a + 58 + \left(50 a + 127\right)\cdot 131 + \left(115 a + 80\right)\cdot 131^{2} + \left(43 a + 79\right)\cdot 131^{3} + \left(90 a + 68\right)\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 80 + 67\cdot 131 + 124\cdot 131^{2} + 83\cdot 131^{3} + 98\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 49 a + 120 + \left(41 a + 28\right)\cdot 131 + \left(99 a + 41\right)\cdot 131^{2} + \left(41 a + 38\right)\cdot 131^{3} + \left(127 a + 8\right)\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 6 + 58\cdot 131 + 43\cdot 131^{2} + 76\cdot 131^{3} + 10\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 82 a + 54 + \left(89 a + 14\right)\cdot 131 + \left(31 a + 4\right)\cdot 131^{2} + \left(89 a + 106\right)\cdot 131^{3} + \left(3 a + 82\right)\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 28 a + 77 + \left(80 a + 96\right)\cdot 131 + \left(15 a + 98\right)\cdot 131^{2} + \left(87 a + 8\right)\cdot 131^{3} + \left(40 a + 124\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.
