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:
| $r_{ 1 }$ |
$=$ |
$ 27 a + 19 + \left(7 a + 12\right)\cdot 61 + \left(46 a + 43\right)\cdot 61^{2} + \left(23 a + 51\right)\cdot 61^{3} + \left(40 a + 13\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 44 + 28\cdot 61 + 52\cdot 61^{2} + 47\cdot 61^{3} + 52\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 50 a + 45 + \left(15 a + 2\right)\cdot 61 + \left(35 a + 25\right)\cdot 61^{2} + \left(34 a + 37\right)\cdot 61^{3} + \left(2 a + 50\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 58 + 55\cdot 61 + 57\cdot 61^{2} + 40\cdot 61^{3} + 16\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 34 a + 46 + \left(53 a + 53\right)\cdot 61 + \left(14 a + 20\right)\cdot 61^{2} + \left(37 a + 29\right)\cdot 61^{3} + \left(20 a + 30\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 11 a + 34 + \left(45 a + 29\right)\cdot 61 + \left(25 a + 44\right)\cdot 61^{2} + \left(26 a + 36\right)\cdot 61^{3} + \left(58 a + 18\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)$ |
| $(2,3)$ |
| $(2,3,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $-2$ |
| $6$ | $2$ | $(2,3)$ | $0$ |
| $9$ | $2$ | $(1,4)(2,3)$ | $0$ |
| $4$ | $3$ | $(2,3,6)$ | $-2$ |
| $4$ | $3$ | $(1,4,5)(2,3,6)$ | $1$ |
| $18$ | $4$ | $(1,2,4,3)(5,6)$ | $0$ |
| $12$ | $6$ | $(1,2,4,3,5,6)$ | $1$ |
| $12$ | $6$ | $(1,4,5)(2,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
