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:
| $r_{ 1 }$ |
$=$ |
$ 55 + 31\cdot 61 + 51\cdot 61^{2} + 5\cdot 61^{3} + 29\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 24 a + 34 + \left(46 a + 17\right)\cdot 61 + \left(5 a + 56\right)\cdot 61^{2} + \left(15 a + 12\right)\cdot 61^{3} + \left(59 a + 29\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 56 a + 10 + \left(11 a + 46\right)\cdot 61 + \left(48 a + 47\right)\cdot 61^{2} + \left(42 a + 56\right)\cdot 61^{3} + \left(6 a + 54\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 37 a + 58 + \left(14 a + 39\right)\cdot 61 + \left(55 a + 15\right)\cdot 61^{2} + \left(45 a + 22\right)\cdot 61^{3} + \left(a + 12\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 5 a + 5 + \left(49 a + 2\right)\cdot 61 + \left(12 a + 23\right)\cdot 61^{2} + \left(18 a + 51\right)\cdot 61^{3} + \left(54 a + 18\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 23 + 45\cdot 61 + 49\cdot 61^{2} + 33\cdot 61^{3} + 38\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,4)$ |
| $(1,2)(3,4)(5,6)$ |
| $(2,4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $2$ |
| $6$ | $2$ | $(2,4)$ | $0$ |
| $9$ | $2$ | $(1,3)(2,4)$ | $0$ |
| $4$ | $3$ | $(1,3,5)(2,4,6)$ | $1$ |
| $4$ | $3$ | $(1,3,5)$ | $-2$ |
| $18$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $12$ | $6$ | $(1,4,3,6,5,2)$ | $-1$ |
| $12$ | $6$ | $(1,3,5)(2,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
