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:
| $r_{ 1 }$ |
$=$ |
$ 36 + 18\cdot 149 + 97\cdot 149^{2} + 144\cdot 149^{3} + 81\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 76 a + 41 + \left(91 a + 129\right)\cdot 149 + \left(85 a + 25\right)\cdot 149^{2} + \left(16 a + 99\right)\cdot 149^{3} + \left(39 a + 45\right)\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 118 a + 71 + \left(142 a + 34\right)\cdot 149 + \left(126 a + 106\right)\cdot 149^{2} + \left(132 a + 115\right)\cdot 149^{3} + \left(18 a + 32\right)\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 73 a + 47 + \left(57 a + 121\right)\cdot 149 + \left(63 a + 127\right)\cdot 149^{2} + \left(132 a + 79\right)\cdot 149^{3} + \left(109 a + 36\right)\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 31 a + 96 + \left(6 a + 40\right)\cdot 149 + \left(22 a + 24\right)\cdot 149^{2} + \left(16 a + 73\right)\cdot 149^{3} + \left(130 a + 124\right)\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 7 + 103\cdot 149 + 65\cdot 149^{2} + 83\cdot 149^{3} + 125\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$5$ |
| $15$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$3$ |
| $15$ |
$2$ |
$(1,2)$ |
$-1$ |
| $45$ |
$2$ |
$(1,2)(3,4)$ |
$1$ |
| $40$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$2$ |
| $40$ |
$3$ |
$(1,2,3)$ |
$-1$ |
| $90$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$-1$ |
| $90$ |
$4$ |
$(1,2,3,4)$ |
$1$ |
| $144$ |
$5$ |
$(1,2,3,4,5)$ |
$0$ |
| $120$ |
$6$ |
$(1,2,3,4,5,6)$ |
$0$ |
| $120$ |
$6$ |
$(1,2,3)(4,5)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
