Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$: $ x^{2} + 60 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 40 + 31\cdot 61 + 8\cdot 61^{2} + 45\cdot 61^{3} + 20\cdot 61^{4} + 10\cdot 61^{5} + 60\cdot 61^{6} + 19\cdot 61^{7} +O\left(61^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 57 + 30\cdot 61 + 37\cdot 61^{2} + 11\cdot 61^{3} + 42\cdot 61^{5} + 33\cdot 61^{6} + 20\cdot 61^{7} +O\left(61^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 16 a + 37 + \left(30 a + 2\right)\cdot 61 + \left(58 a + 38\right)\cdot 61^{2} + \left(10 a + 4\right)\cdot 61^{3} + \left(19 a + 16\right)\cdot 61^{4} + \left(26 a + 23\right)\cdot 61^{5} + \left(20 a + 26\right)\cdot 61^{6} + \left(12 a + 25\right)\cdot 61^{7} +O\left(61^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 45 a + 53 + \left(30 a + 16\right)\cdot 61 + \left(2 a + 5\right)\cdot 61^{2} + \left(50 a + 18\right)\cdot 61^{3} + \left(41 a + 24\right)\cdot 61^{4} + \left(34 a + 30\right)\cdot 61^{5} + \left(40 a + 20\right)\cdot 61^{6} + \left(48 a + 17\right)\cdot 61^{7} +O\left(61^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 41 a + 40 + \left(53 a + 13\right)\cdot 61 + \left(7 a + 39\right)\cdot 61^{2} + \left(58 a + 26\right)\cdot 61^{3} + \left(11 a + 53\right)\cdot 61^{4} + \left(38 a + 55\right)\cdot 61^{5} + \left(8 a + 35\right)\cdot 61^{6} + \left(57 a + 25\right)\cdot 61^{7} +O\left(61^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 20 a + 20 + \left(7 a + 26\right)\cdot 61 + \left(53 a + 54\right)\cdot 61^{2} + \left(2 a + 15\right)\cdot 61^{3} + \left(49 a + 7\right)\cdot 61^{4} + \left(22 a + 21\right)\cdot 61^{5} + \left(52 a + 6\right)\cdot 61^{6} + \left(3 a + 13\right)\cdot 61^{7} +O\left(61^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3,4)(2,5,6)$ |
| $(4,6)$ |
| $(1,4)(2,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,2)(3,5)(4,6)$ | $-3$ |
| $3$ | $2$ | $(3,5)$ | $1$ |
| $3$ | $2$ | $(3,5)(4,6)$ | $-1$ |
| $6$ | $2$ | $(1,4)(2,6)$ | $1$ |
| $6$ | $2$ | $(1,4)(2,6)(3,5)$ | $-1$ |
| $8$ | $3$ | $(1,3,4)(2,5,6)$ | $0$ |
| $6$ | $4$ | $(3,6,5,4)$ | $1$ |
| $6$ | $4$ | $(1,2)(3,6,5,4)$ | $-1$ |
| $8$ | $6$ | $(1,3,6,2,5,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
