Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 191 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 191 }$: $ x^{2} + 190 x + 19 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 54 + 53\cdot 191 + 22\cdot 191^{2} + 92\cdot 191^{3} + 174\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 68 a + 138 + \left(180 a + 122\right)\cdot 191 + \left(56 a + 47\right)\cdot 191^{2} + \left(118 a + 154\right)\cdot 191^{3} + \left(60 a + 100\right)\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 123 a + 15 + \left(10 a + 44\right)\cdot 191 + \left(134 a + 115\right)\cdot 191^{2} + \left(72 a + 24\right)\cdot 191^{3} + \left(130 a + 43\right)\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 98 a + 15 + \left(20 a + 35\right)\cdot 191 + \left(153 a + 143\right)\cdot 191^{2} + \left(154 a + 63\right)\cdot 191^{3} + \left(49 a + 161\right)\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 117 + 116\cdot 191 + 101\cdot 191^{2} + 149\cdot 191^{3} + 149\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 122 + 52\cdot 191 + 58\cdot 191^{2} + 23\cdot 191^{3} + 78\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 93 a + 113 + \left(170 a + 148\right)\cdot 191 + \left(37 a + 84\right)\cdot 191^{2} + \left(36 a + 65\right)\cdot 191^{3} + \left(141 a + 56\right)\cdot 191^{4} +O\left(191^{ 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 |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$14$ |
| $21$ |
$2$ |
$(1,2)$ |
$-4$ |
| $105$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$0$ |
| $105$ |
$2$ |
$(1,2)(3,4)$ |
$2$ |
| $70$ |
$3$ |
$(1,2,3)$ |
$-1$ |
| $280$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$2$ |
| $210$ |
$4$ |
$(1,2,3,4)$ |
$2$ |
| $630$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$0$ |
| $504$ |
$5$ |
$(1,2,3,4,5)$ |
$-1$ |
| $210$ |
$6$ |
$(1,2,3)(4,5)(6,7)$ |
$-1$ |
| $420$ |
$6$ |
$(1,2,3)(4,5)$ |
$-1$ |
| $840$ |
$6$ |
$(1,2,3,4,5,6)$ |
$0$ |
| $720$ |
$7$ |
$(1,2,3,4,5,6,7)$ |
$0$ |
| $504$ |
$10$ |
$(1,2,3,4,5)(6,7)$ |
$1$ |
| $420$ |
$12$ |
$(1,2,3,4)(5,6,7)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
