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:
| $r_{ 1 }$ |
$=$ |
$ 64 a + 3 + \left(85 a + 39\right)\cdot 113 + \left(53 a + 61\right)\cdot 113^{2} + \left(66 a + 56\right)\cdot 113^{3} + \left(37 a + 13\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 112 a + 89 + \left(99 a + 25\right)\cdot 113 + \left(a + 87\right)\cdot 113^{2} + \left(39 a + 52\right)\cdot 113^{3} + \left(101 a + 101\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 74 a + 47 + \left(74 a + 24\right)\cdot 113 + \left(62 a + 6\right)\cdot 113^{2} + \left(35 a + 64\right)\cdot 113^{3} + \left(94 a + 81\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ a + 77 + \left(13 a + 96\right)\cdot 113 + \left(111 a + 9\right)\cdot 113^{2} + \left(73 a + 67\right)\cdot 113^{3} + \left(11 a + 35\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 39 a + 31 + \left(38 a + 55\right)\cdot 113 + \left(50 a + 5\right)\cdot 113^{2} + \left(77 a + 89\right)\cdot 113^{3} + \left(18 a + 47\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 49 a + 93 + \left(27 a + 97\right)\cdot 113 + \left(59 a + 55\right)\cdot 113^{2} + \left(46 a + 9\right)\cdot 113^{3} + \left(75 a + 59\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,5,6,4,3)$ |
| $(1,6)(2,4)(3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,6)(2,4)(3,5)$ |
$-1$ |
$-1$ |
| $1$ |
$3$ |
$(1,5,4)(2,6,3)$ |
$\zeta_{3}$ |
$-\zeta_{3} - 1$ |
| $1$ |
$3$ |
$(1,4,5)(2,3,6)$ |
$-\zeta_{3} - 1$ |
$\zeta_{3}$ |
| $1$ |
$6$ |
$(1,2,5,6,4,3)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
| $1$ |
$6$ |
$(1,3,4,6,5,2)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
