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:
| $r_{ 1 }$ |
$=$ |
$ 148 a + 108 + \left(5 a + 33\right)\cdot 151 + \left(91 a + 103\right)\cdot 151^{2} + \left(100 a + 69\right)\cdot 151^{3} + \left(109 a + 82\right)\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 90 a + 21 + \left(91 a + 142\right)\cdot 151 + \left(6 a + 19\right)\cdot 151^{2} + \left(39 a + 64\right)\cdot 151^{3} + \left(35 a + 125\right)\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 61 a + 50 + \left(59 a + 84\right)\cdot 151 + \left(144 a + 92\right)\cdot 151^{2} + \left(111 a + 135\right)\cdot 151^{3} + \left(115 a + 5\right)\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 3 a + 102 + \left(145 a + 48\right)\cdot 151 + \left(59 a + 128\right)\cdot 151^{2} + \left(50 a + 28\right)\cdot 151^{3} + \left(41 a + 50\right)\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 101 + 14\cdot 151 + 73\cdot 151^{2} + 32\cdot 151^{3} + 81\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 74 + 129\cdot 151 + 35\cdot 151^{2} + 122\cdot 151^{3} + 107\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,3)$ |
| $(1,2)(3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$10$ |
| $45$ |
$2$ |
$(1,2)(3,4)$ |
$-2$ |
| $40$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$1$ |
| $40$ |
$3$ |
$(1,2,3)$ |
$1$ |
| $90$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$0$ |
| $72$ |
$5$ |
$(1,2,3,4,5)$ |
$0$ |
| $72$ |
$5$ |
$(1,3,4,5,2)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
