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:
| $r_{ 1 }$ |
$=$ |
$ 49 a + 8 + \left(46 a + 43\right)\cdot 61 + \left(3 a + 48\right)\cdot 61^{2} + \left(23 a + 48\right)\cdot 61^{3} + \left(29 a + 6\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 a + 57 + \left(14 a + 40\right)\cdot 61 + \left(57 a + 5\right)\cdot 61^{2} + \left(37 a + 7\right)\cdot 61^{3} + \left(31 a + 13\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 11 + 7\cdot 61 + 42\cdot 61^{2} + 44\cdot 61^{3} + 38\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 58 + 37\cdot 61 + 6\cdot 61^{2} + 5\cdot 61^{3} + 41\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 26 a + 43 + 39\cdot 61 + \left(52 a + 44\right)\cdot 61^{2} + \left(22 a + 22\right)\cdot 61^{3} + 22\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 35 a + 8 + \left(60 a + 14\right)\cdot 61 + \left(8 a + 35\right)\cdot 61^{2} + \left(38 a + 54\right)\cdot 61^{3} + \left(60 a + 60\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(2,5)(4,6)$ |
| $(3,5,6)$ |
| $(3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,3)(2,5)(4,6)$ | $-2$ |
| $6$ | $2$ | $(2,4)$ | $0$ |
| $9$ | $2$ | $(2,4)(5,6)$ | $0$ |
| $4$ | $3$ | $(1,2,4)(3,5,6)$ | $1$ |
| $4$ | $3$ | $(1,2,4)$ | $-2$ |
| $18$ | $4$ | $(1,3)(2,6,4,5)$ | $0$ |
| $12$ | $6$ | $(1,5,2,6,4,3)$ | $1$ |
| $12$ | $6$ | $(2,4)(3,5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
