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:
| $r_{ 1 }$ |
$=$ |
$ 77 a + 83 + \left(18 a + 105\right)\cdot 157 + \left(76 a + 108\right)\cdot 157^{2} + \left(13 a + 83\right)\cdot 157^{3} + \left(78 a + 34\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 48 + 137\cdot 157 + 8\cdot 157^{2} + 87\cdot 157^{3} + 72\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 13 a + 58 + \left(118 a + 35\right)\cdot 157 + \left(105 a + 4\right)\cdot 157^{2} + \left(61 a + 9\right)\cdot 157^{3} + \left(121 a + 20\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 80 a + 154 + \left(138 a + 120\right)\cdot 157 + \left(80 a + 156\right)\cdot 157^{2} + \left(143 a + 74\right)\cdot 157^{3} + \left(78 a + 97\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 91 a + 90 + \left(65 a + 3\right)\cdot 157 + \left(136 a + 130\right)\cdot 157^{2} + \left(75 a + 37\right)\cdot 157^{3} + \left(9 a + 90\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 66 a + 74 + \left(91 a + 83\right)\cdot 157 + \left(20 a + 118\right)\cdot 157^{2} + \left(81 a + 123\right)\cdot 157^{3} + \left(147 a + 61\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 144 a + 123 + \left(38 a + 141\right)\cdot 157 + \left(51 a + 100\right)\cdot 157^{2} + \left(95 a + 54\right)\cdot 157^{3} + \left(35 a + 94\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 |
| $1$ | $1$ | $()$ | $35$ |
| $21$ | $2$ | $(1,2)$ | $-5$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $-1$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $70$ | $3$ | $(1,2,3)$ | $-1$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $210$ | $4$ | $(1,2,3,4)$ | $1$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $1$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $-1$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $0$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
