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:
| $r_{ 1 }$ |
$=$ |
$ 19 a + 45 + \left(11 a + 14\right)\cdot 61 + \left(15 a + 3\right)\cdot 61^{2} + \left(22 a + 36\right)\cdot 61^{3} + \left(42 a + 48\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 55 a + 56 + \left(8 a + 20\right)\cdot 61 + \left(9 a + 1\right)\cdot 61^{2} + \left(37 a + 24\right)\cdot 61^{3} + \left(7 a + 17\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 49 + 10\cdot 61 + 46\cdot 61^{2} + 51\cdot 61^{3} + 23\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 42 a + 3 + \left(49 a + 7\right)\cdot 61 + \left(45 a + 7\right)\cdot 61^{2} + \left(38 a + 43\right)\cdot 61^{3} + \left(18 a + 7\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 42 + 32\cdot 61 + 61^{2} + 37\cdot 61^{3} + 36\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 6 a + 50 + \left(52 a + 35\right)\cdot 61 + \left(51 a + 1\right)\cdot 61^{2} + \left(23 a + 52\right)\cdot 61^{3} + \left(53 a + 48\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)$ |
| $(1,3,2,4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $10$ | $2$ | $(1,2)(3,4)(5,6)$ | $-2$ |
| $15$ | $2$ | $(1,5)(4,6)$ | $0$ |
| $20$ | $3$ | $(1,2,6)(3,4,5)$ | $1$ |
| $30$ | $4$ | $(1,6,5,4)$ | $0$ |
| $24$ | $5$ | $(1,4,2,3,6)$ | $-1$ |
| $20$ | $6$ | $(1,3,2,4,6,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
