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:
| $r_{ 1 }$ |
$=$ |
$ 95 + 112\cdot 113 + 10\cdot 113^{2} + 54\cdot 113^{3} + 94\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 97 a + 4 + \left(47 a + 63\right)\cdot 113 + \left(70 a + 76\right)\cdot 113^{2} + \left(107 a + 37\right)\cdot 113^{3} + \left(79 a + 10\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 16 a + 38 + \left(65 a + 88\right)\cdot 113 + \left(42 a + 82\right)\cdot 113^{2} + \left(5 a + 15\right)\cdot 113^{3} + \left(33 a + 71\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 29 + 19\cdot 113 + 77\cdot 113^{3} + 31\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 39 a + 79 + \left(31 a + 28\right)\cdot 113 + \left(77 a + 88\right)\cdot 113^{2} + \left(108 a + 28\right)\cdot 113^{3} + \left(41 a + 94\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 74 a + 95 + \left(81 a + 26\right)\cdot 113 + \left(35 a + 80\right)\cdot 113^{2} + \left(4 a + 12\right)\cdot 113^{3} + \left(71 a + 37\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,3)$ |
| $(1,2)(3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $9$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $40$ | $3$ | $(1,2,3)$ | $0$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $1$ |
| $72$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $72$ | $5$ | $(1,3,4,5,2)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
