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:
| $r_{ 1 }$ |
$=$ |
$ 26 a + 149 + \left(50 a + 42\right)\cdot 157 + \left(60 a + 47\right)\cdot 157^{2} + \left(30 a + 51\right)\cdot 157^{3} + \left(114 a + 99\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 22 + 78\cdot 157 + 104\cdot 157^{2} + 117\cdot 157^{3} + 85\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 63 a + 138 + \left(48 a + 98\right)\cdot 157 + \left(35 a + 102\right)\cdot 157^{2} + \left(94 a + 154\right)\cdot 157^{3} + \left(109 a + 4\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 106 + 68\cdot 157 + 131\cdot 157^{2} + 145\cdot 157^{3} + 10\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 109 + 107\cdot 157 + 26\cdot 157^{2} + 53\cdot 157^{3} + 113\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 131 a + 122 + \left(106 a + 110\right)\cdot 157 + \left(96 a + 141\right)\cdot 157^{2} + \left(126 a + 142\right)\cdot 157^{3} + \left(42 a + 11\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 94 a + 139 + \left(108 a + 120\right)\cdot 157 + \left(121 a + 73\right)\cdot 157^{2} + \left(62 a + 119\right)\cdot 157^{3} + \left(47 a + 144\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 |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$14$ |
| $21$ |
$2$ |
$(1,2)$ |
$-6$ |
| $105$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$-2$ |
| $105$ |
$2$ |
$(1,2)(3,4)$ |
$2$ |
| $70$ |
$3$ |
$(1,2,3)$ |
$2$ |
| $280$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$-1$ |
| $210$ |
$4$ |
$(1,2,3,4)$ |
$0$ |
| $630$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$0$ |
| $504$ |
$5$ |
$(1,2,3,4,5)$ |
$-1$ |
| $210$ |
$6$ |
$(1,2,3)(4,5)(6,7)$ |
$2$ |
| $420$ |
$6$ |
$(1,2,3)(4,5)$ |
$0$ |
| $840$ |
$6$ |
$(1,2,3,4,5,6)$ |
$1$ |
| $720$ |
$7$ |
$(1,2,3,4,5,6,7)$ |
$0$ |
| $504$ |
$10$ |
$(1,2,3,4,5)(6,7)$ |
$-1$ |
| $420$ |
$12$ |
$(1,2,3,4)(5,6,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
