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:
| $r_{ 1 }$ |
$=$ |
$ 188 + 45\cdot 193 + 5\cdot 193^{2} + 143\cdot 193^{3} + 165\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 40 + 28\cdot 193 + 38\cdot 193^{2} + 152\cdot 193^{3} + 166\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 138 a + 69 + \left(177 a + 124\right)\cdot 193 + \left(10 a + 170\right)\cdot 193^{2} + \left(6 a + 11\right)\cdot 193^{3} + \left(5 a + 142\right)\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 154 + 160\cdot 193 + 14\cdot 193^{2} + 52\cdot 193^{3} + 88\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 55 a + 14 + \left(15 a + 164\right)\cdot 193 + \left(182 a + 3\right)\cdot 193^{2} + \left(186 a + 7\right)\cdot 193^{3} + \left(187 a + 141\right)\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 116 + 55\cdot 193 + 153\cdot 193^{2} + 19\cdot 193^{3} + 68\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 |
| $1$ | $1$ | $()$ | $9$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $3$ |
| $15$ | $2$ | $(1,2)$ | $3$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $40$ | $3$ | $(1,2,3)$ | $0$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $1$ |
| $90$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
