Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 113 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 113 }$: $ x^{2} + 101 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 56 a + 111 + \left(40 a + 79\right)\cdot 113 + \left(84 a + 39\right)\cdot 113^{2} + \left(75 a + 107\right)\cdot 113^{3} + \left(13 a + 71\right)\cdot 113^{4} + \left(92 a + 71\right)\cdot 113^{5} +O\left(113^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 57 a + 105 + \left(72 a + 57\right)\cdot 113 + \left(28 a + 107\right)\cdot 113^{2} + \left(37 a + 27\right)\cdot 113^{3} + \left(99 a + 47\right)\cdot 113^{4} + \left(20 a + 33\right)\cdot 113^{5} +O\left(113^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 54 a + 102 + \left(109 a + 27\right)\cdot 113 + \left(89 a + 4\right)\cdot 113^{2} + \left(44 a + 71\right)\cdot 113^{3} + \left(63 a + 57\right)\cdot 113^{4} + \left(31 a + 35\right)\cdot 113^{5} +O\left(113^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 11 a + 22 + \left(20 a + 62\right)\cdot 113 + \left(5 a + 37\right)\cdot 113^{2} + \left(39 a + 27\right)\cdot 113^{3} + \left(108 a + 24\right)\cdot 113^{4} + \left(14 a + 1\right)\cdot 113^{5} +O\left(113^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 59 a + 72 + \left(3 a + 44\right)\cdot 113 + \left(23 a + 70\right)\cdot 113^{2} + \left(68 a + 66\right)\cdot 113^{3} + \left(49 a + 95\right)\cdot 113^{4} + \left(81 a + 11\right)\cdot 113^{5} +O\left(113^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 102 a + 41 + \left(92 a + 66\right)\cdot 113 + \left(107 a + 79\right)\cdot 113^{2} + \left(73 a + 38\right)\cdot 113^{3} + \left(4 a + 42\right)\cdot 113^{4} + \left(98 a + 72\right)\cdot 113^{5} +O\left(113^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,4,3)$ |
| $(1,4,6,3,5,2)$ |
| $(1,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,3)(2,6)(4,5)$ | $0$ |
| $1$ | $3$ | $(1,6,5)(2,4,3)$ | $-2 \zeta_{3} - 2$ |
| $1$ | $3$ | $(1,5,6)(2,3,4)$ | $2 \zeta_{3}$ |
| $2$ | $3$ | $(2,4,3)$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(2,3,4)$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(1,6,5)(2,3,4)$ | $-1$ |
| $3$ | $6$ | $(1,4,6,3,5,2)$ | $0$ |
| $3$ | $6$ | $(1,2,5,3,6,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
