Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 149 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 149 }$: $ x^{2} + 145 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 120 a + 78 + \left(116 a + 116\right)\cdot 149 + \left(95 a + 80\right)\cdot 149^{2} + \left(129 a + 45\right)\cdot 149^{3} + 106 a\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 142 a + 15 + \left(80 a + 10\right)\cdot 149 + \left(53 a + 132\right)\cdot 149^{2} + \left(54 a + 128\right)\cdot 149^{3} + \left(131 a + 144\right)\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 7 a + 136 + \left(68 a + 42\right)\cdot 149 + \left(95 a + 116\right)\cdot 149^{2} + \left(94 a + 143\right)\cdot 149^{3} + \left(17 a + 19\right)\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 29 a + 111 + \left(32 a + 16\right)\cdot 149 + \left(53 a + 49\right)\cdot 149^{2} + \left(19 a + 21\right)\cdot 149^{3} + 42 a\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 20 a + 21 + \left(25 a + 128\right)\cdot 149 + \left(126 a + 9\right)\cdot 149^{2} + \left(78 a + 127\right)\cdot 149^{3} + \left(31 a + 100\right)\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 129 a + 101 + \left(123 a + 59\right)\cdot 149 + \left(22 a + 42\right)\cdot 149^{2} + \left(70 a + 18\right)\cdot 149^{3} + \left(117 a + 148\right)\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 136 + 72\cdot 149 + 16\cdot 149^{2} + 111\cdot 149^{3} + 32\cdot 149^{4} +O\left(149^{ 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.
