Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 181 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 181 }$: $ x^{2} + 177 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 7 + 156\cdot 181 + 139\cdot 181^{2} + 8\cdot 181^{3} + 138\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 96 a + 173 + \left(125 a + 54\right)\cdot 181 + \left(162 a + 138\right)\cdot 181^{2} + \left(78 a + 16\right)\cdot 181^{3} + \left(165 a + 175\right)\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 90 a + 173 + \left(95 a + 109\right)\cdot 181 + \left(121 a + 179\right)\cdot 181^{2} + \left(77 a + 36\right)\cdot 181^{3} + \left(29 a + 22\right)\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 87 + 168\cdot 181 + 177\cdot 181^{2} + 148\cdot 181^{3} + 59\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 100 + 95\cdot 181 + 121\cdot 181^{2} + 116\cdot 181^{3} + 51\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 85 a + 14 + \left(55 a + 99\right)\cdot 181 + \left(18 a + 120\right)\cdot 181^{2} + \left(102 a + 169\right)\cdot 181^{3} + \left(15 a + 33\right)\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 91 a + 171 + \left(85 a + 39\right)\cdot 181 + \left(59 a + 27\right)\cdot 181^{2} + \left(103 a + 45\right)\cdot 181^{3} + \left(151 a + 62\right)\cdot 181^{4} +O\left(181^{ 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.
