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:
| $r_{ 1 }$ |
$=$ |
$ 88 a + 47 + \left(126 a + 8\right)\cdot 131 + \left(76 a + 46\right)\cdot 131^{2} + \left(83 a + 40\right)\cdot 131^{3} + \left(77 a + 32\right)\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 a + 10 + \left(52 a + 85\right)\cdot 131 + \left(33 a + 105\right)\cdot 131^{2} + \left(97 a + 125\right)\cdot 131^{3} + \left(2 a + 71\right)\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 43 a + 6 + \left(4 a + 34\right)\cdot 131 + \left(54 a + 96\right)\cdot 131^{2} + \left(47 a + 35\right)\cdot 131^{3} + \left(53 a + 128\right)\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 39 + 40\cdot 131 + 92\cdot 131^{2} + 56\cdot 131^{3} + 46\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 50 + 50\cdot 131 + 29\cdot 131^{2} + 110\cdot 131^{3} + 119\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 89 + 15\cdot 131 + 98\cdot 131^{2} + 66\cdot 131^{3} + 8\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 128 a + 22 + \left(78 a + 28\right)\cdot 131 + \left(97 a + 56\right)\cdot 131^{2} + \left(33 a + 88\right)\cdot 131^{3} + \left(128 a + 116\right)\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 values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$15$ |
| $21$ |
$2$ |
$(1,2)$ |
$-5$ |
| $105$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$3$ |
| $105$ |
$2$ |
$(1,2)(3,4)$ |
$-1$ |
| $70$ |
$3$ |
$(1,2,3)$ |
$3$ |
| $280$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$0$ |
| $210$ |
$4$ |
$(1,2,3,4)$ |
$-1$ |
| $630$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$-1$ |
| $504$ |
$5$ |
$(1,2,3,4,5)$ |
$0$ |
| $210$ |
$6$ |
$(1,2,3)(4,5)(6,7)$ |
$-1$ |
| $420$ |
$6$ |
$(1,2,3)(4,5)$ |
$1$ |
| $840$ |
$6$ |
$(1,2,3,4,5,6)$ |
$0$ |
| $720$ |
$7$ |
$(1,2,3,4,5,6,7)$ |
$1$ |
| $504$ |
$10$ |
$(1,2,3,4,5)(6,7)$ |
$0$ |
| $420$ |
$12$ |
$(1,2,3,4)(5,6,7)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
