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:
| $r_{ 1 }$ |
$=$ |
$ 92 a + 45 + \left(107 a + 114\right)\cdot 193 + \left(54 a + 103\right)\cdot 193^{2} + \left(109 a + 89\right)\cdot 193^{3} + \left(102 a + 158\right)\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 102 a + 136 + \left(59 a + 130\right)\cdot 193 + \left(87 a + 96\right)\cdot 193^{2} + \left(18 a + 157\right)\cdot 193^{3} + \left(77 a + 10\right)\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 91 a + 45 + \left(133 a + 88\right)\cdot 193 + \left(105 a + 124\right)\cdot 193^{2} + \left(174 a + 88\right)\cdot 193^{3} + \left(115 a + 69\right)\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 101 + 172\cdot 193 + 32\cdot 193^{2} + 49\cdot 193^{3} + 167\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 101 a + 137 + \left(85 a + 129\right)\cdot 193 + \left(138 a + 50\right)\cdot 193^{2} + \left(83 a + 144\right)\cdot 193^{3} + \left(90 a + 151\right)\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 116 + 136\cdot 193 + 170\cdot 193^{2} + 49\cdot 193^{3} + 21\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.
