Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$: $ x^{2} + 60 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 8 a + 46 + \left(8 a + 56\right)\cdot 61 + \left(19 a + 31\right)\cdot 61^{2} + \left(37 a + 28\right)\cdot 61^{3} + \left(53 a + 22\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 22 + 8\cdot 61 + 47\cdot 61^{2} + 46\cdot 61^{3} + 60\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 9 a + 32 + \left(31 a + 5\right)\cdot 61 + \left(23 a + 10\right)\cdot 61^{2} + \left(30 a + 26\right)\cdot 61^{3} + \left(55 a + 14\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 49 + 27\cdot 61 + 48\cdot 61^{2} + 61^{3} + 7\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 53 a + 54 + \left(52 a + 56\right)\cdot 61 + \left(41 a + 42\right)\cdot 61^{2} + \left(23 a + 46\right)\cdot 61^{3} + \left(7 a + 38\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 52 a + 41 + \left(29 a + 27\right)\cdot 61 + \left(37 a + 2\right)\cdot 61^{2} + \left(30 a + 33\right)\cdot 61^{3} + \left(5 a + 39\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(2,4)(5,6)$ |
| $(3,4)$ |
| $(3,4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,3)(2,4)(5,6)$ | $2$ |
| $6$ | $2$ | $(3,4)$ | $0$ |
| $9$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $4$ | $3$ | $(1,2,5)(3,4,6)$ | $1$ |
| $4$ | $3$ | $(1,2,5)$ | $-2$ |
| $18$ | $4$ | $(1,3,2,4)(5,6)$ | $0$ |
| $12$ | $6$ | $(1,4,2,6,5,3)$ | $-1$ |
| $12$ | $6$ | $(1,2,5)(3,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
