Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 103 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 103 }$: $ x^{2} + 102 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 25 a + 30 + \left(93 a + 95\right)\cdot 103 + \left(60 a + 5\right)\cdot 103^{2} + \left(102 a + 99\right)\cdot 103^{3} + \left(27 a + 65\right)\cdot 103^{4} + \left(83 a + 17\right)\cdot 103^{5} + \left(15 a + 93\right)\cdot 103^{6} +O\left(103^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 63 a + 84 + \left(85 a + 32\right)\cdot 103 + \left(44 a + 52\right)\cdot 103^{2} + \left(27 a + 7\right)\cdot 103^{3} + \left(5 a + 71\right)\cdot 103^{4} + \left(102 a + 56\right)\cdot 103^{5} + \left(26 a + 62\right)\cdot 103^{6} +O\left(103^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 78 a + 55 + \left(9 a + 60\right)\cdot 103 + \left(42 a + 76\right)\cdot 103^{2} + 37\cdot 103^{3} + \left(75 a + 94\right)\cdot 103^{4} + \left(19 a + 72\right)\cdot 103^{5} + \left(87 a + 25\right)\cdot 103^{6} +O\left(103^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 40 a + 44 + \left(17 a + 55\right)\cdot 103 + \left(58 a + 11\right)\cdot 103^{2} + \left(75 a + 93\right)\cdot 103^{3} + \left(97 a + 48\right)\cdot 103^{4} + 50\cdot 103^{5} + \left(76 a + 90\right)\cdot 103^{6} +O\left(103^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 86 a + 57 + \left(44 a + 1\right)\cdot 103 + \left(37 a + 85\right)\cdot 103^{2} + \left(10 a + 100\right)\cdot 103^{3} + \left(60 a + 40\right)\cdot 103^{4} + \left(97 a + 88\right)\cdot 103^{5} + \left(95 a + 70\right)\cdot 103^{6} +O\left(103^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 17 a + 40 + \left(58 a + 63\right)\cdot 103 + \left(65 a + 77\right)\cdot 103^{2} + \left(92 a + 73\right)\cdot 103^{3} + \left(42 a + 90\right)\cdot 103^{4} + \left(5 a + 22\right)\cdot 103^{5} + \left(7 a + 69\right)\cdot 103^{6} +O\left(103^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4,6)$ |
| $(1,3)(2,4)(5,6)$ |
| $(1,6,4)(2,5,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,3)(2,4)(5,6)$ | $0$ |
| $1$ | $3$ | $(1,4,6)(2,5,3)$ | $-2 \zeta_{3} - 2$ |
| $1$ | $3$ | $(1,6,4)(2,3,5)$ | $2 \zeta_{3}$ |
| $2$ | $3$ | $(1,6,4)(2,5,3)$ | $-1$ |
| $2$ | $3$ | $(1,4,6)$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(1,6,4)$ | $\zeta_{3} + 1$ |
| $3$ | $6$ | $(1,3,4,2,6,5)$ | $0$ |
| $3$ | $6$ | $(1,5,6,2,4,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
