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:
| $r_{ 1 }$ |
$=$ |
$ 8 + 53\cdot 61 + 13\cdot 61^{2} + 16\cdot 61^{3} + 45\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 26 a + 48 + \left(17 a + 34\right)\cdot 61 + \left(41 a + 18\right)\cdot 61^{2} + \left(5 a + 48\right)\cdot 61^{3} + 2\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 50 a + 36 + \left(15 a + 47\right)\cdot 61 + \left(52 a + 42\right)\cdot 61^{2} + \left(15 a + 48\right)\cdot 61^{3} + \left(47 a + 14\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 53 + 7\cdot 61 + 47\cdot 61^{2} + 44\cdot 61^{3} + 15\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 35 a + 13 + \left(43 a + 26\right)\cdot 61 + \left(19 a + 42\right)\cdot 61^{2} + \left(55 a + 12\right)\cdot 61^{3} + \left(60 a + 58\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 11 a + 25 + \left(45 a + 13\right)\cdot 61 + \left(8 a + 18\right)\cdot 61^{2} + \left(45 a + 12\right)\cdot 61^{3} + \left(13 a + 46\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,5)(3,6)$ |
| $(1,3,5)(2,4,6)$ |
| $(1,6,2)(3,5,4)$ |
| $(1,2)(3,6)(4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(2,5)(3,6)$ | $-1$ |
| $6$ | $2$ | $(1,2)(3,6)(4,5)$ | $1$ |
| $8$ | $3$ | $(1,3,5)(2,4,6)$ | $0$ |
| $6$ | $4$ | $(1,5,4,2)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
