Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 113 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 113 }$: $ x^{2} + 101 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 43 a + 91 + \left(91 a + 36\right)\cdot 113 + \left(38 a + 94\right)\cdot 113^{2} + \left(70 a + 45\right)\cdot 113^{3} + \left(59 a + 67\right)\cdot 113^{4} + \left(70 a + 23\right)\cdot 113^{5} + \left(93 a + 92\right)\cdot 113^{6} +O\left(113^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 36 a + 100 + \left(33 a + 13\right)\cdot 113 + \left(71 a + 60\right)\cdot 113^{2} + \left(38 a + 76\right)\cdot 113^{3} + \left(26 a + 102\right)\cdot 113^{4} + \left(55 a + 51\right)\cdot 113^{5} + \left(47 a + 36\right)\cdot 113^{6} +O\left(113^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 70 a + 42 + \left(21 a + 73\right)\cdot 113 + \left(74 a + 16\right)\cdot 113^{2} + \left(42 a + 60\right)\cdot 113^{3} + \left(53 a + 34\right)\cdot 113^{4} + \left(42 a + 19\right)\cdot 113^{5} + \left(19 a + 15\right)\cdot 113^{6} +O\left(113^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 77 a + 80 + \left(79 a + 38\right)\cdot 113 + \left(41 a + 91\right)\cdot 113^{2} + \left(74 a + 16\right)\cdot 113^{3} + \left(86 a + 41\right)\cdot 113^{4} + \left(57 a + 10\right)\cdot 113^{5} + \left(65 a + 99\right)\cdot 113^{6} +O\left(113^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 107 + 77\cdot 113 + 101\cdot 113^{2} + 93\cdot 113^{3} + 31\cdot 113^{4} + 75\cdot 113^{5} + 77\cdot 113^{6} +O\left(113^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 33 + 98\cdot 113 + 87\cdot 113^{2} + 45\cdot 113^{3} + 61\cdot 113^{4} + 45\cdot 113^{5} + 18\cdot 113^{6} +O\left(113^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)$ |
| $(1,2,5)(3,6,4)$ |
| $(1,2)(3,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,4)(2,3)(5,6)$ | $-3$ |
| $3$ | $2$ | $(1,4)$ | $1$ |
| $3$ | $2$ | $(1,4)(2,3)$ | $-1$ |
| $6$ | $2$ | $(2,5)(3,6)$ | $1$ |
| $6$ | $2$ | $(1,4)(2,5)(3,6)$ | $-1$ |
| $8$ | $3$ | $(1,2,5)(3,6,4)$ | $0$ |
| $6$ | $4$ | $(1,3,4,2)$ | $1$ |
| $6$ | $4$ | $(1,4)(2,6,3,5)$ | $-1$ |
| $8$ | $6$ | $(1,3,6,4,2,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
