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 + 152 + \left(119 a + 131\right)\cdot 157 + \left(34 a + 89\right)\cdot 157^{2} + \left(123 a + 101\right)\cdot 157^{3} + \left(112 a + 63\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 53 a + 74 + \left(62 a + 109\right)\cdot 157 + \left(5 a + 122\right)\cdot 157^{2} + \left(80 a + 49\right)\cdot 157^{3} + \left(81 a + 67\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 134 a + 31 + \left(82 a + 25\right)\cdot 157 + \left(152 a + 20\right)\cdot 157^{2} + \left(124 a + 156\right)\cdot 157^{3} + \left(133 a + 125\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 104 a + 25 + \left(94 a + 54\right)\cdot 157 + \left(151 a + 87\right)\cdot 157^{2} + \left(76 a + 130\right)\cdot 157^{3} + \left(75 a + 80\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 58 + 102\cdot 157 + 91\cdot 157^{2} + 134\cdot 157^{3} + 57\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 23 a + 73 + \left(74 a + 148\right)\cdot 157 + \left(4 a + 71\right)\cdot 157^{2} + 32 a\cdot 157^{3} + \left(23 a + 42\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 113 a + 58 + \left(37 a + 56\right)\cdot 157 + \left(122 a + 144\right)\cdot 157^{2} + \left(33 a + 54\right)\cdot 157^{3} + \left(44 a + 33\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 values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$15$ |
| $21$ |
$2$ |
$(1,2)$ |
$5$ |
| $105$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$-3$ |
| $105$ |
$2$ |
$(1,2)(3,4)$ |
$-1$ |
| $70$ |
$3$ |
$(1,2,3)$ |
$3$ |
| $280$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$0$ |
| $210$ |
$4$ |
$(1,2,3,4)$ |
$1$ |
| $630$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$-1$ |
| $504$ |
$5$ |
$(1,2,3,4,5)$ |
$0$ |
| $210$ |
$6$ |
$(1,2,3)(4,5)(6,7)$ |
$-1$ |
| $420$ |
$6$ |
$(1,2,3)(4,5)$ |
$-1$ |
| $840$ |
$6$ |
$(1,2,3,4,5,6)$ |
$0$ |
| $720$ |
$7$ |
$(1,2,3,4,5,6,7)$ |
$1$ |
| $504$ |
$10$ |
$(1,2,3,4,5)(6,7)$ |
$0$ |
| $420$ |
$12$ |
$(1,2,3,4)(5,6,7)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
