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:
| $r_{ 1 }$ |
$=$ |
$ 73 a + 5 + \left(94 a + 92\right)\cdot 149 + \left(102 a + 42\right)\cdot 149^{2} + \left(59 a + 102\right)\cdot 149^{3} + \left(97 a + 60\right)\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 64 + 60\cdot 149 + 102\cdot 149^{2} + 41\cdot 149^{3} + 106\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 70 a + 48 + \left(43 a + 84\right)\cdot 149 + \left(80 a + 138\right)\cdot 149^{2} + \left(20 a + 38\right)\cdot 149^{3} + \left(52 a + 4\right)\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 76 a + 148 + \left(54 a + 98\right)\cdot 149 + \left(46 a + 60\right)\cdot 149^{2} + \left(89 a + 89\right)\cdot 149^{3} + \left(51 a + 92\right)\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 79 a + 30 + \left(105 a + 39\right)\cdot 149 + \left(68 a + 118\right)\cdot 149^{2} + \left(128 a + 40\right)\cdot 149^{3} + \left(96 a + 43\right)\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 98 + 116\cdot 149 + 78\cdot 149^{2} + 121\cdot 149^{3} + 120\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 56 + 104\cdot 149 + 54\cdot 149^{2} + 12\cdot 149^{3} + 19\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 |
| $1$ | $1$ | $()$ | $15$ |
| $21$ | $2$ | $(1,2)$ | $5$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $-3$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $70$ | $3$ | $(1,2,3)$ | $3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $210$ | $4$ | $(1,2,3,4)$ | $1$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $1$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $0$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
