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:
| $r_{ 1 }$ |
$=$ |
$ 124 a + 100 + \left(134 a + 114\right)\cdot 157 + \left(88 a + 120\right)\cdot 157^{2} + \left(82 a + 7\right)\cdot 157^{3} + \left(88 a + 141\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 149 a + 98 + \left(155 a + 110\right)\cdot 157 + \left(143 a + 56\right)\cdot 157^{2} + \left(41 a + 78\right)\cdot 157^{3} + \left(80 a + 109\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 8 a + 58 + \left(a + 113\right)\cdot 157 + \left(13 a + 149\right)\cdot 157^{2} + \left(115 a + 143\right)\cdot 157^{3} + \left(76 a + 154\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 123 + 95\cdot 157 + 27\cdot 157^{2} + 66\cdot 157^{3} + 35\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 33 a + 92 + \left(22 a + 36\right)\cdot 157 + \left(68 a + 116\right)\cdot 157^{2} + \left(74 a + 17\right)\cdot 157^{3} + \left(68 a + 30\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$5$ |
| $10$ |
$2$ |
$(1,2)$ |
$1$ |
| $15$ |
$2$ |
$(1,2)(3,4)$ |
$1$ |
| $20$ |
$3$ |
$(1,2,3)$ |
$-1$ |
| $30$ |
$4$ |
$(1,2,3,4)$ |
$-1$ |
| $24$ |
$5$ |
$(1,2,3,4,5)$ |
$0$ |
| $20$ |
$6$ |
$(1,2,3)(4,5)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
