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:
| $r_{ 1 }$ |
$=$ |
$ 75 + 41\cdot 113 + 108\cdot 113^{3} + 100\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 83 a + 18 + \left(7 a + 4\right)\cdot 113 + \left(69 a + 93\right)\cdot 113^{2} + \left(a + 62\right)\cdot 113^{3} + \left(39 a + 112\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 107 + 94\cdot 113 + 45\cdot 113^{2} + 26\cdot 113^{3} + 28\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 30 a + 110 + \left(105 a + 13\right)\cdot 113 + \left(43 a + 10\right)\cdot 113^{2} + \left(111 a + 13\right)\cdot 113^{3} + \left(73 a + 14\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 20 a + 64 + \left(20 a + 37\right)\cdot 113 + \left(107 a + 83\right)\cdot 113^{2} + \left(64 a + 10\right)\cdot 113^{3} + \left(37 a + 18\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 93 a + 78 + \left(92 a + 33\right)\cdot 113 + \left(5 a + 106\right)\cdot 113^{2} + \left(48 a + 4\right)\cdot 113^{3} + \left(75 a + 65\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.
