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:
| $r_{ 1 }$ |
$=$ |
$ 19 a + 21 + \left(44 a + 48\right)\cdot 61 + \left(46 a + 59\right)\cdot 61^{2} + \left(25 a + 40\right)\cdot 61^{3} + \left(a + 42\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 13 a + 24 + \left(53 a + 10\right)\cdot 61 + \left(54 a + 60\right)\cdot 61^{2} + \left(30 a + 11\right)\cdot 61^{3} + \left(47 a + 22\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 10 + 49\cdot 61 + 58\cdot 61^{2} + 44\cdot 61^{3} + 3\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 42 a + 40 + \left(16 a + 12\right)\cdot 61 + \left(14 a + 1\right)\cdot 61^{2} + \left(35 a + 20\right)\cdot 61^{3} + \left(59 a + 18\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 48 a + 37 + \left(7 a + 50\right)\cdot 61 + 6 a\cdot 61^{2} + \left(30 a + 49\right)\cdot 61^{3} + \left(13 a + 38\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 51 + 11\cdot 61 + 2\cdot 61^{2} + 16\cdot 61^{3} + 57\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,5)(3,6)$ |
| $(1,6,4,3)$ |
| $(1,5,3)(2,6,4)$ |
| $(1,3,2)(4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(2,5)(3,6)$ | $-1$ |
| $6$ | $2$ | $(1,5)(2,4)(3,6)$ | $1$ |
| $8$ | $3$ | $(1,3,2)(4,6,5)$ | $0$ |
| $6$ | $4$ | $(1,3,4,6)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
