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:
| $r_{ 1 }$ |
$=$ |
$ 33 a + 17 + \left(181 a + 9\right)\cdot 193 + \left(57 a + 51\right)\cdot 193^{2} + \left(26 a + 151\right)\cdot 193^{3} + \left(159 a + 52\right)\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 161 a + 78 + \left(5 a + 112\right)\cdot 193 + \left(5 a + 104\right)\cdot 193^{2} + \left(68 a + 160\right)\cdot 193^{3} + \left(143 a + 192\right)\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 142 a + 124 + \left(91 a + 3\right)\cdot 193 + \left(145 a + 169\right)\cdot 193^{2} + \left(15 a + 26\right)\cdot 193^{3} + \left(74 a + 7\right)\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 160 a + 50 + \left(11 a + 157\right)\cdot 193 + \left(135 a + 120\right)\cdot 193^{2} + \left(166 a + 119\right)\cdot 193^{3} + \left(33 a + 185\right)\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 32 a + 46 + \left(187 a + 150\right)\cdot 193 + \left(187 a + 103\right)\cdot 193^{2} + \left(124 a + 30\right)\cdot 193^{3} + \left(49 a + 75\right)\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 51 a + 73 + \left(101 a + 146\right)\cdot 193 + \left(47 a + 29\right)\cdot 193^{2} + \left(177 a + 90\right)\cdot 193^{3} + \left(118 a + 65\right)\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 |
| $1$ | $1$ | $()$ | $5$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $3$ |
| $15$ | $2$ | $(1,2)$ | $-1$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $2$ |
| $40$ | $3$ | $(1,2,3)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)$ | $1$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
