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:
| $r_{ 1 }$ |
$=$ |
$ 167 a + 52 + \left(186 a + 78\right)\cdot 193 + \left(190 a + 25\right)\cdot 193^{2} + \left(157 a + 90\right)\cdot 193^{3} + \left(112 a + 99\right)\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 49 + 23\cdot 193 + 35\cdot 193^{2} + 151\cdot 193^{3} + 93\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 26 a + 26 + \left(6 a + 98\right)\cdot 193 + \left(2 a + 29\right)\cdot 193^{2} + \left(35 a + 57\right)\cdot 193^{3} + \left(80 a + 54\right)\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 62 + 104\cdot 193 + 147\cdot 193^{2} + 13\cdot 193^{3} + 32\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 134 a + 128 + \left(57 a + 175\right)\cdot 193 + \left(141 a + 128\right)\cdot 193^{2} + \left(69 a + 72\right)\cdot 193^{3} + \left(149 a + 13\right)\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 59 a + 69 + \left(135 a + 99\right)\cdot 193 + \left(51 a + 19\right)\cdot 193^{2} + \left(123 a + 1\right)\cdot 193^{3} + \left(43 a + 93\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.
