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:
| $r_{ 1 }$ |
$=$ |
$ 102 a + 4 + \left(126 a + 81\right)\cdot 149 + \left(113 a + 123\right)\cdot 149^{2} + \left(88 a + 112\right)\cdot 149^{3} + \left(125 a + 8\right)\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 47 a + 114 + \left(22 a + 38\right)\cdot 149 + \left(35 a + 5\right)\cdot 149^{2} + \left(60 a + 56\right)\cdot 149^{3} + \left(23 a + 124\right)\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 87 a + 64 + \left(103 a + 96\right)\cdot 149 + \left(57 a + 98\right)\cdot 149^{2} + \left(62 a + 136\right)\cdot 149^{3} + \left(125 a + 85\right)\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 62 a + 114 + \left(45 a + 125\right)\cdot 149 + \left(91 a + 76\right)\cdot 149^{2} + \left(86 a + 30\right)\cdot 149^{3} + \left(23 a + 78\right)\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 104 + 57\cdot 149 + 6\cdot 149^{2} + 90\cdot 149^{3} + 85\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 47 + 47\cdot 149 + 136\cdot 149^{2} + 20\cdot 149^{3} + 64\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 |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$16$ |
| $15$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$0$ |
| $15$ |
$2$ |
$(1,2)$ |
$0$ |
| $45$ |
$2$ |
$(1,2)(3,4)$ |
$0$ |
| $40$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$-2$ |
| $40$ |
$3$ |
$(1,2,3)$ |
$-2$ |
| $90$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$0$ |
| $90$ |
$4$ |
$(1,2,3,4)$ |
$0$ |
| $144$ |
$5$ |
$(1,2,3,4,5)$ |
$1$ |
| $120$ |
$6$ |
$(1,2,3,4,5,6)$ |
$0$ |
| $120$ |
$6$ |
$(1,2,3)(4,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
