Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 193 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 193 }$: $ x^{2} + 192 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 72 a + 150 + \left(125 a + 111\right)\cdot 193 + \left(136 a + 152\right)\cdot 193^{2} + \left(142 a + 150\right)\cdot 193^{3} + \left(28 a + 76\right)\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 143 a + 31 + \left(136 a + 42\right)\cdot 193 + \left(66 a + 90\right)\cdot 193^{2} + \left(191 a + 105\right)\cdot 193^{3} + \left(171 a + 181\right)\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 35 + 33\cdot 193 + 130\cdot 193^{2} + 193^{3} + 48\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 121 a + 29 + \left(67 a + 165\right)\cdot 193 + \left(56 a + 163\right)\cdot 193^{2} + \left(50 a + 156\right)\cdot 193^{3} + \left(164 a + 155\right)\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 50 a + 174 + \left(56 a + 35\right)\cdot 193 + \left(126 a + 20\right)\cdot 193^{2} + \left(a + 37\right)\cdot 193^{3} + \left(21 a + 162\right)\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 162 + 190\cdot 193 + 21\cdot 193^{2} + 127\cdot 193^{3} + 147\cdot 193^{4} +O\left(193^{ 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 |
| $1$ | $1$ | $()$ | $5$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $1$ |
| $15$ | $2$ | $(1,2)$ | $-3$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $40$ | $3$ | $(1,2,3)$ | $2$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $1$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
