Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$: $ x^{2} + 60 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 57 a + 26 + \left(43 a + 51\right)\cdot 61 + \left(29 a + 51\right)\cdot 61^{2} + \left(32 a + 50\right)\cdot 61^{3} + \left(27 a + 47\right)\cdot 61^{4} + \left(13 a + 34\right)\cdot 61^{5} + \left(58 a + 16\right)\cdot 61^{6} +O\left(61^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 58 a + 42 + \left(29 a + 32\right)\cdot 61 + \left(40 a + 46\right)\cdot 61^{2} + \left(37 a + 55\right)\cdot 61^{3} + \left(9 a + 1\right)\cdot 61^{4} + \left(53 a + 32\right)\cdot 61^{5} + \left(14 a + 57\right)\cdot 61^{6} +O\left(61^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 4 a + 22 + \left(17 a + 38\right)\cdot 61 + \left(31 a + 37\right)\cdot 61^{2} + \left(28 a + 53\right)\cdot 61^{3} + \left(33 a + 42\right)\cdot 61^{4} + \left(47 a + 20\right)\cdot 61^{5} + 2 a\cdot 61^{6} +O\left(61^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 14 + 32\cdot 61 + 32\cdot 61^{2} + 17\cdot 61^{3} + 31\cdot 61^{4} + 5\cdot 61^{5} + 44\cdot 61^{6} +O\left(61^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 42 + 23\cdot 61 + 18\cdot 61^{2} + 13\cdot 61^{3} + 24\cdot 61^{4} + 14\cdot 61^{5} + 45\cdot 61^{6} +O\left(61^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 3 a + 39 + \left(31 a + 4\right)\cdot 61 + \left(20 a + 57\right)\cdot 61^{2} + \left(23 a + 52\right)\cdot 61^{3} + \left(51 a + 34\right)\cdot 61^{4} + \left(7 a + 14\right)\cdot 61^{5} + \left(46 a + 19\right)\cdot 61^{6} +O\left(61^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(3,4)(5,6)$ |
| $(1,2)(3,6)(4,5)$ |
| $(1,3)(2,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,6)(4,5)$ | $-2$ |
| $3$ | $2$ | $(1,3)(2,6)$ | $0$ |
| $3$ | $2$ | $(1,6)(2,3)(4,5)$ | $0$ |
| $2$ | $3$ | $(1,4,3)(2,5,6)$ | $-1$ |
| $2$ | $6$ | $(1,5,3,2,4,6)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
