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:
| $r_{ 1 }$ |
$=$ |
$ 149 a + 43 + \left(142 a + 58\right)\cdot 181 + \left(64 a + 90\right)\cdot 181^{2} + \left(49 a + 148\right)\cdot 181^{3} + \left(27 a + 123\right)\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 32 a + 96 + \left(38 a + 118\right)\cdot 181 + \left(116 a + 25\right)\cdot 181^{2} + \left(131 a + 100\right)\cdot 181^{3} + \left(153 a + 2\right)\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 168 a + 113 + \left(64 a + 45\right)\cdot 181 + \left(176 a + 41\right)\cdot 181^{2} + \left(96 a + 83\right)\cdot 181^{3} + \left(46 a + 151\right)\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 60 a + 101 + \left(91 a + 119\right)\cdot 181 + \left(92 a + 33\right)\cdot 181^{2} + \left(169 a + 166\right)\cdot 181^{3} + \left(157 a + 66\right)\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 151 + 82\cdot 181^{2} + 84\cdot 181^{3} + 152\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 13 a + 61 + \left(116 a + 137\right)\cdot 181 + \left(4 a + 138\right)\cdot 181^{2} + \left(84 a + 113\right)\cdot 181^{3} + \left(134 a + 59\right)\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 121 a + 160 + \left(89 a + 62\right)\cdot 181 + \left(88 a + 131\right)\cdot 181^{2} + \left(11 a + 27\right)\cdot 181^{3} + \left(23 a + 167\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 value |
| $1$ | $1$ | $()$ | $15$ |
| $21$ | $2$ | $(1,2)$ | $5$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $-3$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $70$ | $3$ | $(1,2,3)$ | $3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $210$ | $4$ | $(1,2,3,4)$ | $1$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $1$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $0$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
