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:
| $r_{ 1 }$ |
$=$ |
$ 126 a + 70 + \left(128 a + 126\right)\cdot 181 + \left(12 a + 77\right)\cdot 181^{2} + \left(177 a + 73\right)\cdot 181^{3} + \left(130 a + 150\right)\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 172 + 110\cdot 181 + 37\cdot 181^{2} + 135\cdot 181^{3} + 23\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 61 a + 92 + \left(7 a + 51\right)\cdot 181 + \left(29 a + 92\right)\cdot 181^{2} + \left(111 a + 118\right)\cdot 181^{3} + \left(90 a + 114\right)\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 152 + 135\cdot 181 + 154\cdot 181^{2} + 113\cdot 181^{3} + 149\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 120 a + 155 + \left(173 a + 19\right)\cdot 181 + \left(151 a + 20\right)\cdot 181^{2} + \left(69 a + 172\right)\cdot 181^{3} + \left(90 a + 3\right)\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 55 a + 31 + \left(52 a + 153\right)\cdot 181 + \left(168 a + 180\right)\cdot 181^{2} + \left(3 a + 44\right)\cdot 181^{3} + \left(50 a + 135\right)\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 54 + 126\cdot 181 + 160\cdot 181^{2} + 65\cdot 181^{3} + 146\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 |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$6$ |
| $21$ |
$2$ |
$(1,2)$ |
$4$ |
| $105$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$0$ |
| $105$ |
$2$ |
$(1,2)(3,4)$ |
$2$ |
| $70$ |
$3$ |
$(1,2,3)$ |
$3$ |
| $280$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$0$ |
| $210$ |
$4$ |
$(1,2,3,4)$ |
$2$ |
| $630$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$0$ |
| $504$ |
$5$ |
$(1,2,3,4,5)$ |
$1$ |
| $210$ |
$6$ |
$(1,2,3)(4,5)(6,7)$ |
$-1$ |
| $420$ |
$6$ |
$(1,2,3)(4,5)$ |
$1$ |
| $840$ |
$6$ |
$(1,2,3,4,5,6)$ |
$0$ |
| $720$ |
$7$ |
$(1,2,3,4,5,6,7)$ |
$-1$ |
| $504$ |
$10$ |
$(1,2,3,4,5)(6,7)$ |
$-1$ |
| $420$ |
$12$ |
$(1,2,3,4)(5,6,7)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
