Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 107 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 107 }$: $ x^{2} + 103 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 66 a + 14 + \left(106 a + 37\right)\cdot 107 + \left(48 a + 50\right)\cdot 107^{2} + \left(29 a + 17\right)\cdot 107^{3} + \left(90 a + 8\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 41 a + 64 + 76\cdot 107 + \left(58 a + 32\right)\cdot 107^{2} + \left(77 a + 86\right)\cdot 107^{3} + \left(16 a + 18\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 72 a + 65 + \left(44 a + 68\right)\cdot 107 + \left(86 a + 69\right)\cdot 107^{2} + \left(93 a + 89\right)\cdot 107^{3} + \left(52 a + 45\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 35 a + 32 + \left(62 a + 68\right)\cdot 107 + \left(20 a + 49\right)\cdot 107^{2} + \left(13 a + 57\right)\cdot 107^{3} + \left(54 a + 56\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 85 + 81\cdot 107 + 58\cdot 107^{2} + 17\cdot 107^{3} + 6\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 63 + 95\cdot 107 + 59\cdot 107^{2} + 52\cdot 107^{3} + 78\cdot 107^{4} +O\left(107^{ 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.
