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:
| $r_{ 1 }$ |
$=$ |
$ 25 a + 18 + \left(58 a + 44\right)\cdot 61 + \left(28 a + 14\right)\cdot 61^{2} + \left(49 a + 20\right)\cdot 61^{3} + \left(56 a + 57\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 38 + 19\cdot 61 + 57\cdot 61^{2} + 19\cdot 61^{3} + 43\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 28 a + 47 + \left(46 a + 51\right)\cdot 61 + \left(10 a + 17\right)\cdot 61^{2} + \left(55 a + 8\right)\cdot 61^{3} + \left(28 a + 13\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 36 a + 43 + \left(2 a + 16\right)\cdot 61 + \left(32 a + 46\right)\cdot 61^{2} + \left(11 a + 40\right)\cdot 61^{3} + \left(4 a + 3\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 23 + 41\cdot 61 + 3\cdot 61^{2} + 41\cdot 61^{3} + 17\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 33 a + 14 + \left(14 a + 9\right)\cdot 61 + \left(50 a + 43\right)\cdot 61^{2} + \left(5 a + 52\right)\cdot 61^{3} + \left(32 a + 47\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,5)(3,6)$ |
| $(1,3)(2,5)(4,6)$ |
| $(1,3,2)(4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(2,5)(3,6)$ | $-1$ |
| $6$ | $2$ | $(1,3)(2,5)(4,6)$ | $1$ |
| $8$ | $3$ | $(1,3,2)(4,6,5)$ | $0$ |
| $6$ | $4$ | $(1,3,4,6)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
