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:
| $r_{ 1 }$ |
$=$ |
$ 87 a + 38 + \left(143 a + 25\right)\cdot 157 + \left(42 a + 98\right)\cdot 157^{2} + \left(112 a + 11\right)\cdot 157^{3} + \left(112 a + 73\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 70 a + 2 + \left(13 a + 28\right)\cdot 157 + \left(114 a + 12\right)\cdot 157^{2} + \left(44 a + 59\right)\cdot 157^{3} + \left(44 a + 53\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 143 a + 72 + \left(81 a + 47\right)\cdot 157 + \left(24 a + 71\right)\cdot 157^{2} + \left(88 a + 16\right)\cdot 157^{3} + \left(3 a + 131\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 14 a + 2 + 75 a\cdot 157 + \left(132 a + 112\right)\cdot 157^{2} + \left(68 a + 118\right)\cdot 157^{3} + \left(153 a + 60\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 36 a + 90 + \left(29 a + 51\right)\cdot 157 + \left(21 a + 50\right)\cdot 157^{2} + \left(49 a + 20\right)\cdot 157^{3} + \left(136 a + 74\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 121 a + 113 + \left(127 a + 4\right)\cdot 157 + \left(135 a + 127\right)\cdot 157^{2} + \left(107 a + 87\right)\cdot 157^{3} + \left(20 a + 78\right)\cdot 157^{4} +O\left(157^{ 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 value |
| $1$ | $1$ | $()$ | $5$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $3$ |
| $15$ | $2$ | $(1,2)$ | $-1$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $2$ |
| $40$ | $3$ | $(1,2,3)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)$ | $1$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
