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:
| $r_{ 1 }$ |
$=$ |
$ 67 + 139\cdot 193 + 134\cdot 193^{2} + 101\cdot 193^{3} + 74\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 98 + 189\cdot 193 + 100\cdot 193^{2} + 118\cdot 193^{3} + 21\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 176 a + 155 + \left(141 a + 115\right)\cdot 193 + \left(138 a + 43\right)\cdot 193^{2} + \left(172 a + 146\right)\cdot 193^{3} + \left(105 a + 112\right)\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 98 a + 12 + \left(186 a + 175\right)\cdot 193 + \left(140 a + 55\right)\cdot 193^{2} + \left(98 a + 37\right)\cdot 193^{3} + \left(59 a + 85\right)\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 95 a + 110 + \left(6 a + 70\right)\cdot 193 + \left(52 a + 10\right)\cdot 193^{2} + \left(94 a + 188\right)\cdot 193^{3} + \left(133 a + 45\right)\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 17 a + 138 + \left(51 a + 81\right)\cdot 193 + \left(54 a + 40\right)\cdot 193^{2} + \left(20 a + 180\right)\cdot 193^{3} + \left(87 a + 45\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 values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$9$ |
| $15$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$-3$ |
| $15$ |
$2$ |
$(1,2)$ |
$-3$ |
| $45$ |
$2$ |
$(1,2)(3,4)$ |
$1$ |
| $40$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$0$ |
| $40$ |
$3$ |
$(1,2,3)$ |
$0$ |
| $90$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$1$ |
| $90$ |
$4$ |
$(1,2,3,4)$ |
$1$ |
| $144$ |
$5$ |
$(1,2,3,4,5)$ |
$-1$ |
| $120$ |
$6$ |
$(1,2,3,4,5,6)$ |
$0$ |
| $120$ |
$6$ |
$(1,2,3)(4,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
