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:
| $r_{ 1 }$ |
$=$ |
$ 58 a + 55 + \left(47 a + 62\right)\cdot 113 + \left(31 a + 88\right)\cdot 113^{2} + \left(22 a + 44\right)\cdot 113^{3} + \left(23 a + 48\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 24 + 5\cdot 113 + 9\cdot 113^{2} + 63\cdot 113^{3} + 41\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 95 a + 21 + \left(104 a + 33\right)\cdot 113 + \left(99 a + 92\right)\cdot 113^{2} + \left(12 a + 25\right)\cdot 113^{3} + \left(72 a + 98\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 55 a + 73 + \left(65 a + 9\right)\cdot 113 + \left(81 a + 79\right)\cdot 113^{2} + \left(90 a + 54\right)\cdot 113^{3} + \left(89 a + 78\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 18 a + 31 + \left(8 a + 66\right)\cdot 113 + \left(13 a + 56\right)\cdot 113^{2} + \left(100 a + 80\right)\cdot 113^{3} + \left(40 a + 46\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 22 + 49\cdot 113 + 13\cdot 113^{2} + 70\cdot 113^{3} + 25\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 |
| $1$ | $1$ | $()$ | $8$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $40$ | $3$ | $(1,2,3)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $72$ | $5$ | $(1,2,3,4,5)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
| $72$ | $5$ | $(1,3,4,5,2)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.
