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:
| $r_{ 1 }$ |
$=$ |
$ 54 a + 19 + \left(8 a + 55\right)\cdot 61 + \left(35 a + 27\right)\cdot 61^{2} + \left(37 a + 45\right)\cdot 61^{3} + \left(4 a + 49\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 30 + 28\cdot 61 + 39\cdot 61^{2} + 57\cdot 61^{3} + 50\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 43 + 25\cdot 61 + 43\cdot 61^{2} + 21\cdot 61^{3} + 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 7 a + 12 + \left(52 a + 10\right)\cdot 61 + \left(25 a + 54\right)\cdot 61^{2} + \left(23 a + 47\right)\cdot 61^{3} + \left(56 a + 16\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 59 a + 42 + \left(11 a + 24\right)\cdot 61 + 29 a\cdot 61^{2} + \left(52 a + 24\right)\cdot 61^{3} + \left(50 a + 2\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 2 a + 40 + \left(49 a + 38\right)\cdot 61 + \left(31 a + 17\right)\cdot 61^{2} + \left(8 a + 47\right)\cdot 61^{3} + 10 a\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4,6,5,3,2)$ |
| $(1,2)(3,6)(4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $10$ | $2$ | $(1,2)(3,6)(4,5)$ | $-2$ |
| $15$ | $2$ | $(1,2)(4,6)$ | $0$ |
| $20$ | $3$ | $(1,6,3)(2,4,5)$ | $1$ |
| $30$ | $4$ | $(1,4,2,6)$ | $0$ |
| $24$ | $5$ | $(1,3,4,6,5)$ | $-1$ |
| $20$ | $6$ | $(1,4,6,5,3,2)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
