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:
| $r_{ 1 }$ |
$=$ |
$ 107 a + 64 + \left(98 a + 25\right)\cdot 113 + \left(37 a + 54\right)\cdot 113^{2} + \left(55 a + 87\right)\cdot 113^{3} + \left(63 a + 38\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 18 + 49\cdot 113 + 74\cdot 113^{2} + 76\cdot 113^{3} + 99\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 6 a + 105 + \left(14 a + 88\right)\cdot 113 + \left(75 a + 70\right)\cdot 113^{2} + \left(57 a + 35\right)\cdot 113^{3} + \left(49 a + 67\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 10 a + 18 + \left(49 a + 26\right)\cdot 113 + \left(54 a + 32\right)\cdot 113^{2} + \left(44 a + 103\right)\cdot 113^{3} + \left(62 a + 75\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 109 + 108\cdot 113 + 35\cdot 113^{2} + 18\cdot 113^{3} + 68\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 103 a + 25 + \left(63 a + 40\right)\cdot 113 + \left(58 a + 71\right)\cdot 113^{2} + \left(68 a + 17\right)\cdot 113^{3} + \left(50 a + 102\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 values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$8$ |
$8$ |
| $45$ |
$2$ |
$(1,2)(3,4)$ |
$0$ |
$0$ |
| $40$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$-1$ |
$-1$ |
| $40$ |
$3$ |
$(1,2,3)$ |
$-1$ |
$-1$ |
| $90$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$0$ |
$0$ |
| $72$ |
$5$ |
$(1,2,3,4,5)$ |
$\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
| $72$ |
$5$ |
$(1,3,4,5,2)$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
$\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
