Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 157 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 157 }$: $ x^{2} + 152 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 44 a + 15 + \left(60 a + 110\right)\cdot 157 + \left(101 a + 131\right)\cdot 157^{2} + \left(43 a + 27\right)\cdot 157^{3} + \left(77 a + 58\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 91 a + 98 + \left(145 a + 58\right)\cdot 157 + \left(137 a + 147\right)\cdot 157^{2} + \left(83 a + 83\right)\cdot 157^{3} + \left(36 a + 129\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 66 a + 82 + \left(11 a + 67\right)\cdot 157 + \left(19 a + 63\right)\cdot 157^{2} + \left(73 a + 51\right)\cdot 157^{3} + \left(120 a + 71\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 113 a + 78 + \left(96 a + 53\right)\cdot 157 + \left(55 a + 107\right)\cdot 157^{2} + \left(113 a + 144\right)\cdot 157^{3} + \left(79 a + 86\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 113 + 29\cdot 157 + 54\cdot 157^{2} + 28\cdot 157^{3} + 32\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 108 a + 9 + \left(82 a + 80\right)\cdot 157 + \left(135 a + 78\right)\cdot 157^{2} + \left(55 a + 152\right)\cdot 157^{3} + \left(58 a + 6\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 49 a + 78 + \left(74 a + 71\right)\cdot 157 + \left(21 a + 45\right)\cdot 157^{2} + \left(101 a + 139\right)\cdot 157^{3} + \left(98 a + 85\right)\cdot 157^{4} +O\left(157^{ 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 |
| $1$ | $1$ | $()$ | $14$ |
| $21$ | $2$ | $(1,2)$ | $-4$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $2$ |
| $70$ | $3$ | $(1,2,3)$ | $-1$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $2$ |
| $210$ | $4$ | $(1,2,3,4)$ | $2$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
