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:
| $r_{ 1 }$ |
$=$ |
$ 48 + 39\cdot 61 + 18\cdot 61^{2} + 6\cdot 61^{3} + 19\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 60 a + 7 + \left(21 a + 60\right)\cdot 61 + \left(53 a + 35\right)\cdot 61^{2} + \left(19 a + 13\right)\cdot 61^{3} + \left(59 a + 1\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 29 + 5\cdot 61 + 15\cdot 61^{2} + 14\cdot 61^{3} + 57\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 17 a + 38 + \left(11 a + 30\right)\cdot 61 + \left(51 a + 33\right)\cdot 61^{2} + \left(41 a + 58\right)\cdot 61^{3} + \left(45 a + 60\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ a + 6 + \left(39 a + 22\right)\cdot 61 + \left(7 a + 6\right)\cdot 61^{2} + \left(41 a + 41\right)\cdot 61^{3} + \left(a + 40\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 44 a + 55 + \left(49 a + 24\right)\cdot 61 + \left(9 a + 12\right)\cdot 61^{2} + \left(19 a + 49\right)\cdot 61^{3} + \left(15 a + 3\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(2,4)(5,6)$ |
| $(3,4)$ |
| $(3,4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,3)(2,4)(5,6)$ | $-2$ |
| $6$ | $2$ | $(2,5)$ | $0$ |
| $9$ | $2$ | $(2,5)(4,6)$ | $0$ |
| $4$ | $3$ | $(1,2,5)(3,4,6)$ | $1$ |
| $4$ | $3$ | $(1,2,5)$ | $-2$ |
| $18$ | $4$ | $(1,3)(2,6,5,4)$ | $0$ |
| $12$ | $6$ | $(1,4,2,6,5,3)$ | $1$ |
| $12$ | $6$ | $(2,5)(3,4,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
