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:
| $r_{ 1 }$ |
$=$ |
$ 87 + 184\cdot 193 + 162\cdot 193^{2} + 168\cdot 193^{3} + 36\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 90 a + 121 + \left(79 a + 19\right)\cdot 193 + \left(184 a + 65\right)\cdot 193^{2} + \left(54 a + 153\right)\cdot 193^{3} + \left(101 a + 147\right)\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 10 a + 159 + \left(82 a + 101\right)\cdot 193 + \left(170 a + 59\right)\cdot 193^{2} + \left(115 a + 108\right)\cdot 193^{3} + \left(50 a + 151\right)\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 103 a + 18 + \left(113 a + 9\right)\cdot 193 + \left(8 a + 170\right)\cdot 193^{2} + \left(138 a + 23\right)\cdot 193^{3} + \left(91 a + 1\right)\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 28 + 90\cdot 193 + 166\cdot 193^{2} + 70\cdot 193^{3} + 155\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 183 a + 169 + \left(110 a + 173\right)\cdot 193 + \left(22 a + 147\right)\cdot 193^{2} + \left(77 a + 53\right)\cdot 193^{3} + \left(142 a + 86\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 |
| $1$ | $1$ | $()$ | $10$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-2$ |
| $15$ | $2$ | $(1,2)$ | $2$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $-2$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $1$ |
| $40$ | $3$ | $(1,2,3)$ | $1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $90$ | $4$ | $(1,2,3,4)$ | $0$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $1$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
