Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$: $ x^{2} + 60 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 25 a + 33 + \left(49 a + 47\right)\cdot 61 + \left(50 a + 8\right)\cdot 61^{2} + \left(31 a + 17\right)\cdot 61^{3} + \left(31 a + 47\right)\cdot 61^{4} + \left(13 a + 33\right)\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 36 a + 58 + \left(11 a + 10\right)\cdot 61 + \left(10 a + 10\right)\cdot 61^{2} + \left(29 a + 59\right)\cdot 61^{3} + \left(29 a + 46\right)\cdot 61^{4} + \left(47 a + 15\right)\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 47 + 26\cdot 61 + 57\cdot 61^{2} + 21\cdot 61^{3} + 15\cdot 61^{4} + 58\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 25 a + 4 + \left(49 a + 50\right)\cdot 61 + \left(50 a + 50\right)\cdot 61^{2} + \left(31 a + 1\right)\cdot 61^{3} + \left(31 a + 14\right)\cdot 61^{4} + \left(13 a + 45\right)\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 15 + 34\cdot 61 + 3\cdot 61^{2} + 39\cdot 61^{3} + 45\cdot 61^{4} + 2\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 36 a + 29 + \left(11 a + 13\right)\cdot 61 + \left(10 a + 52\right)\cdot 61^{2} + \left(29 a + 43\right)\cdot 61^{3} + \left(29 a + 13\right)\cdot 61^{4} + \left(47 a + 27\right)\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(4,6)$ |
| $(1,6)$ |
| $(1,2,3)(4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,6)(2,4)(3,5)$ | $-3$ |
| $3$ | $2$ | $(1,6)$ | $1$ |
| $3$ | $2$ | $(1,6)(2,4)$ | $-1$ |
| $6$ | $2$ | $(2,3)(4,5)$ | $-1$ |
| $6$ | $2$ | $(1,6)(2,3)(4,5)$ | $1$ |
| $8$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $6$ | $4$ | $(1,4,6,2)$ | $-1$ |
| $6$ | $4$ | $(1,6)(2,5,4,3)$ | $1$ |
| $8$ | $6$ | $(1,4,5,6,2,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
