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:
| $r_{ 1 }$ |
$=$ |
$ 67 + 68\cdot 103 + 77\cdot 103^{2} + 99\cdot 103^{3} + 14\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 46 a + 64 + \left(83 a + 46\right)\cdot 103 + \left(84 a + 32\right)\cdot 103^{2} + \left(8 a + 39\right)\cdot 103^{3} + \left(18 a + 71\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 99 + 95\cdot 103 + 15\cdot 103^{2} + 6\cdot 103^{3} + 69\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 8 a + 84 + \left(89 a + 17\right)\cdot 103 + \left(85 a + 76\right)\cdot 103^{2} + \left(81 a + 50\right)\cdot 103^{3} + \left(7 a + 73\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 57 a + 7 + \left(19 a + 84\right)\cdot 103 + \left(18 a + 33\right)\cdot 103^{2} + \left(94 a + 66\right)\cdot 103^{3} + \left(84 a + 80\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 95 a + 92 + \left(13 a + 98\right)\cdot 103 + \left(17 a + 72\right)\cdot 103^{2} + \left(21 a + 46\right)\cdot 103^{3} + \left(95 a + 102\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 |
| $1$ | $1$ | $()$ | $5$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $3$ |
| $15$ | $2$ | $(1,2)$ | $-1$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $2$ |
| $40$ | $3$ | $(1,2,3)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)$ | $1$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
