Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 151 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 151 }$: $ x^{2} + 149 x + 6 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 22 a + \left(115 a + 18\right)\cdot 151 + \left(147 a + 18\right)\cdot 151^{2} + \left(102 a + 130\right)\cdot 151^{3} + 135\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 129 a + 44 + \left(35 a + 75\right)\cdot 151 + \left(3 a + 47\right)\cdot 151^{2} + \left(48 a + 37\right)\cdot 151^{3} + \left(150 a + 34\right)\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 101 a + 146 + \left(137 a + 88\right)\cdot 151 + \left(2 a + 125\right)\cdot 151^{2} + \left(124 a + 77\right)\cdot 151^{3} + \left(77 a + 16\right)\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 84 a + 129 + \left(30 a + 76\right)\cdot 151 + \left(89 a + 86\right)\cdot 151^{2} + \left(124 a + 22\right)\cdot 151^{3} + \left(104 a + 15\right)\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 50 a + 46 + \left(13 a + 112\right)\cdot 151 + \left(148 a + 144\right)\cdot 151^{2} + \left(26 a + 20\right)\cdot 151^{3} + \left(73 a + 48\right)\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 94 + 27\cdot 151 + 98\cdot 151^{2} + 132\cdot 151^{3} + 102\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 67 a + 146 + \left(120 a + 53\right)\cdot 151 + \left(61 a + 83\right)\cdot 151^{2} + \left(26 a + 31\right)\cdot 151^{3} + \left(46 a + 100\right)\cdot 151^{4} +O\left(151^{ 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 |
| $1$ | $1$ | $()$ | $15$ |
| $21$ | $2$ | $(1,2)$ | $-5$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $3$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $70$ | $3$ | $(1,2,3)$ | $3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $210$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $1$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $0$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
