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:
| $r_{ 1 }$ |
$=$ |
$ 119 + 95\cdot 157 + 44\cdot 157^{2} + 72\cdot 157^{3} + 99\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 148 a + 40 + \left(95 a + 38\right)\cdot 157 + \left(94 a + 121\right)\cdot 157^{2} + \left(106 a + 72\right)\cdot 157^{3} + \left(131 a + 97\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 59 + 63\cdot 157 + 5\cdot 157^{2} + 131\cdot 157^{3} + 46\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 25 a + 145 + \left(88 a + 57\right)\cdot 157 + \left(107 a + 68\right)\cdot 157^{2} + \left(82 a + 81\right)\cdot 157^{3} + \left(114 a + 93\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 9 a + 152 + \left(61 a + 55\right)\cdot 157 + \left(62 a + 27\right)\cdot 157^{2} + \left(50 a + 40\right)\cdot 157^{3} + \left(25 a + 21\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 132 a + 113 + \left(68 a + 2\right)\cdot 157 + \left(49 a + 47\right)\cdot 157^{2} + \left(74 a + 73\right)\cdot 157^{3} + \left(42 a + 112\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 values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$10$ |
| $15$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$-2$ |
| $15$ |
$2$ |
$(1,2)$ |
$2$ |
| $45$ |
$2$ |
$(1,2)(3,4)$ |
$-2$ |
| $40$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$1$ |
| $40$ |
$3$ |
$(1,2,3)$ |
$1$ |
| $90$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$0$ |
| $90$ |
$4$ |
$(1,2,3,4)$ |
$0$ |
| $144$ |
$5$ |
$(1,2,3,4,5)$ |
$0$ |
| $120$ |
$6$ |
$(1,2,3,4,5,6)$ |
$1$ |
| $120$ |
$6$ |
$(1,2,3)(4,5)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
