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:
| $r_{ 1 }$ |
$=$ |
$ 23 a + 18 + \left(190 a + 60\right)\cdot 193 + \left(181 a + 46\right)\cdot 193^{2} + \left(105 a + 42\right)\cdot 193^{3} + \left(13 a + 160\right)\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 57 a + 120 + \left(112 a + 18\right)\cdot 193 + \left(104 a + 55\right)\cdot 193^{2} + \left(125 a + 189\right)\cdot 193^{3} + \left(90 a + 151\right)\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 177 + 7\cdot 193 + 84\cdot 193^{2} + 133\cdot 193^{3} + 169\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 169 + 183\cdot 193 + 61\cdot 193^{2} + 130\cdot 193^{3} +O\left(193^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 170 a + 41 + \left(2 a + 34\right)\cdot 193 + \left(11 a + 38\right)\cdot 193^{2} + \left(87 a + 159\right)\cdot 193^{3} + \left(179 a + 67\right)\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 71 + 7\cdot 193 + 53\cdot 193^{2} + 100\cdot 193^{3} + 104\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 136 a + 177 + \left(80 a + 73\right)\cdot 193 + \left(88 a + 47\right)\cdot 193^{2} + \left(67 a + 17\right)\cdot 193^{3} + \left(102 a + 117\right)\cdot 193^{4} +O\left(193^{ 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 value |
| $1$ | $1$ | $()$ | $14$ |
| $21$ | $2$ | $(1,2)$ | $-4$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $2$ |
| $70$ | $3$ | $(1,2,3)$ | $-1$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $2$ |
| $210$ | $4$ | $(1,2,3,4)$ | $2$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
