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:
| $r_{ 1 }$ |
$=$ |
$ 98 a + 183 + \left(61 a + 66\right)\cdot 191 + \left(127 a + 123\right)\cdot 191^{2} + \left(49 a + 32\right)\cdot 191^{3} + \left(44 a + 122\right)\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 183 a + \left(25 a + 87\right)\cdot 191 + \left(105 a + 67\right)\cdot 191^{2} + \left(20 a + 165\right)\cdot 191^{3} + \left(101 a + 38\right)\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 8 a + 183 + \left(165 a + 120\right)\cdot 191 + \left(85 a + 146\right)\cdot 191^{2} + \left(170 a + 80\right)\cdot 191^{3} + \left(89 a + 119\right)\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 188 a + 131 + \left(108 a + 105\right)\cdot 191 + \left(181 a + 2\right)\cdot 191^{2} + \left(72 a + 37\right)\cdot 191^{3} + \left(152 a + 115\right)\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 3 a + 128 + \left(82 a + 26\right)\cdot 191 + \left(9 a + 75\right)\cdot 191^{2} + \left(118 a + 119\right)\cdot 191^{3} + \left(38 a + 3\right)\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 49 + 135\cdot 191 + 159\cdot 191^{2} + 182\cdot 191^{3} + 56\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 93 a + 90 + \left(129 a + 30\right)\cdot 191 + \left(63 a + 189\right)\cdot 191^{2} + \left(141 a + 145\right)\cdot 191^{3} + \left(146 a + 116\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 |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$15$ |
| $21$ |
$2$ |
$(1,2)$ |
$5$ |
| $105$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$-3$ |
| $105$ |
$2$ |
$(1,2)(3,4)$ |
$-1$ |
| $70$ |
$3$ |
$(1,2,3)$ |
$3$ |
| $280$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$0$ |
| $210$ |
$4$ |
$(1,2,3,4)$ |
$1$ |
| $630$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$-1$ |
| $504$ |
$5$ |
$(1,2,3,4,5)$ |
$0$ |
| $210$ |
$6$ |
$(1,2,3)(4,5)(6,7)$ |
$-1$ |
| $420$ |
$6$ |
$(1,2,3)(4,5)$ |
$-1$ |
| $840$ |
$6$ |
$(1,2,3,4,5,6)$ |
$0$ |
| $720$ |
$7$ |
$(1,2,3,4,5,6,7)$ |
$1$ |
| $504$ |
$10$ |
$(1,2,3,4,5)(6,7)$ |
$0$ |
| $420$ |
$12$ |
$(1,2,3,4)(5,6,7)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
