Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 163 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 163 }$: $ x^{2} + 159 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 46 + 120\cdot 163 + 63\cdot 163^{2} + 62\cdot 163^{3} + 5\cdot 163^{4} +O\left(163^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 100 a + 25 + \left(71 a + 81\right)\cdot 163 + \left(39 a + 131\right)\cdot 163^{2} + \left(2 a + 129\right)\cdot 163^{3} + \left(66 a + 60\right)\cdot 163^{4} +O\left(163^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 38 a + 23 + \left(162 a + 69\right)\cdot 163 + \left(76 a + 150\right)\cdot 163^{2} + \left(49 a + 65\right)\cdot 163^{3} + \left(132 a + 143\right)\cdot 163^{4} +O\left(163^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 109 + 86\cdot 163 + 9\cdot 163^{2} + 100\cdot 163^{3} + 126\cdot 163^{4} +O\left(163^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 12 + 162\cdot 163 + 108\cdot 163^{2} + 7\cdot 163^{3} + 22\cdot 163^{4} +O\left(163^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 125 a + 12 + 28\cdot 163 + \left(86 a + 133\right)\cdot 163^{2} + \left(113 a + 23\right)\cdot 163^{3} + \left(30 a + 134\right)\cdot 163^{4} +O\left(163^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 63 a + 99 + \left(91 a + 104\right)\cdot 163 + \left(123 a + 54\right)\cdot 163^{2} + \left(160 a + 99\right)\cdot 163^{3} + \left(96 a + 159\right)\cdot 163^{4} +O\left(163^{ 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 |
| $1$ | $1$ | $()$ | $21$ |
| $21$ | $2$ | $(1,2)$ | $1$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $-3$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $70$ | $3$ | $(1,2,3)$ | $-3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $210$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
