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:
| $r_{ 1 }$ |
$=$ |
$ 32 + 21\cdot 103 + 58\cdot 103^{2} + 95\cdot 103^{3} + 23\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 65 a + 3 + \left(58 a + 76\right)\cdot 103 + \left(5 a + 17\right)\cdot 103^{2} + \left(34 a + 98\right)\cdot 103^{3} + \left(11 a + 65\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 56 a + 37 + \left(39 a + 100\right)\cdot 103 + \left(41 a + 2\right)\cdot 103^{2} + \left(54 a + 80\right)\cdot 103^{3} + \left(21 a + 29\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 77 + 60\cdot 103 + 54\cdot 103^{2} + 21\cdot 103^{3} + 46\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 38 a + 68 + \left(44 a + 69\right)\cdot 103 + \left(97 a + 67\right)\cdot 103^{2} + \left(68 a + 23\right)\cdot 103^{3} + \left(91 a + 43\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 47 a + 93 + \left(63 a + 83\right)\cdot 103 + \left(61 a + 4\right)\cdot 103^{2} + \left(48 a + 93\right)\cdot 103^{3} + \left(81 a + 99\right)\cdot 103^{4} +O\left(103^{ 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.
