Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 103 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 103 }$: $ x^{2} + 102 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 96 a + 6 + \left(6 a + 28\right)\cdot 103 + \left(95 a + 47\right)\cdot 103^{2} + \left(20 a + 11\right)\cdot 103^{3} + \left(90 a + 4\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 15 a + 14 + \left(53 a + 9\right)\cdot 103 + \left(39 a + 59\right)\cdot 103^{2} + \left(19 a + 69\right)\cdot 103^{3} + \left(38 a + 30\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 88 a + 29 + \left(49 a + 47\right)\cdot 103 + \left(63 a + 45\right)\cdot 103^{2} + \left(83 a + 49\right)\cdot 103^{3} + \left(64 a + 49\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 55 + 79\cdot 103 + 21\cdot 103^{2} + 35\cdot 103^{3} + 48\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 7 a + 102 + \left(96 a + 41\right)\cdot 103 + \left(7 a + 32\right)\cdot 103^{2} + \left(82 a + 40\right)\cdot 103^{3} + \left(12 a + 73\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character value |
| $1$ | $1$ | $()$ | $5$ |
| $10$ | $2$ | $(1,2)$ | $-1$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $20$ | $3$ | $(1,2,3)$ | $-1$ |
| $30$ | $4$ | $(1,2,3,4)$ | $1$ |
| $24$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
