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 value |
| $1$ | $1$ | $()$ | $35$ |
| $21$ | $2$ | $(1,2)$ | $5$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $1$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $70$ | $3$ | $(1,2,3)$ | $-1$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $210$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $1$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $1$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $0$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
