Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 113 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 113 }$: $ x^{2} + 101 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 36 + 34\cdot 113^{2} + 19\cdot 113^{3} + 11\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 75 + 23\cdot 113 + 75\cdot 113^{2} + 103\cdot 113^{3} + 55\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 65 a + 95 + \left(85 a + 30\right)\cdot 113 + \left(80 a + 5\right)\cdot 113^{2} + \left(72 a + 3\right)\cdot 113^{3} + \left(17 a + 29\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 11 a + 15 + \left(52 a + 73\right)\cdot 113 + \left(98 a + 63\right)\cdot 113^{2} + \left(4 a + 11\right)\cdot 113^{3} + \left(112 a + 102\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 48 a + 84 + \left(27 a + 88\right)\cdot 113 + \left(32 a + 97\right)\cdot 113^{2} + \left(40 a + 3\right)\cdot 113^{3} + \left(95 a + 55\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 102 a + 34 + \left(60 a + 9\right)\cdot 113 + \left(14 a + 63\right)\cdot 113^{2} + \left(108 a + 84\right)\cdot 113^{3} + 85\cdot 113^{4} +O\left(113^{ 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.
