Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 13.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$: $ x^{2} + 60 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 27\cdot 61 + 53\cdot 61^{2} + 6\cdot 61^{3} + 14\cdot 61^{5} + 13\cdot 61^{6} + 26\cdot 61^{7} + 13\cdot 61^{8} + 7\cdot 61^{9} + 7\cdot 61^{10} + 47\cdot 61^{11} + 58\cdot 61^{12} +O\left(61^{ 13 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 57 a + 2 + \left(31 a + 43\right)\cdot 61 + 15\cdot 61^{2} + \left(56 a + 33\right)\cdot 61^{3} + \left(19 a + 48\right)\cdot 61^{4} + \left(25 a + 27\right)\cdot 61^{5} + \left(11 a + 37\right)\cdot 61^{6} + \left(56 a + 38\right)\cdot 61^{7} + \left(23 a + 46\right)\cdot 61^{8} + \left(28 a + 58\right)\cdot 61^{9} + \left(59 a + 14\right)\cdot 61^{10} + \left(13 a + 53\right)\cdot 61^{11} + \left(4 a + 4\right)\cdot 61^{12} +O\left(61^{ 13 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 26 a + 48 + \left(43 a + 21\right)\cdot 61 + \left(25 a + 39\right)\cdot 61^{2} + \left(13 a + 36\right)\cdot 61^{3} + \left(49 a + 12\right)\cdot 61^{4} + \left(18 a + 15\right)\cdot 61^{5} + \left(51 a + 14\right)\cdot 61^{6} + \left(4 a + 23\right)\cdot 61^{7} + \left(34 a + 46\right)\cdot 61^{8} + \left(51 a + 21\right)\cdot 61^{9} + \left(49 a + 31\right)\cdot 61^{10} + \left(49 a + 30\right)\cdot 61^{11} + \left(34 a + 7\right)\cdot 61^{12} +O\left(61^{ 13 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 19 + 27\cdot 61 + 20\cdot 61^{2} + 31\cdot 61^{3} + 45\cdot 61^{4} + 5\cdot 61^{5} + 6\cdot 61^{6} + 17\cdot 61^{7} + 37\cdot 61^{8} + 32\cdot 61^{9} + 39\cdot 61^{10} + 36\cdot 61^{11} +O\left(61^{ 13 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 60 + 33\cdot 61 + 7\cdot 61^{2} + 54\cdot 61^{3} + 60\cdot 61^{4} + 46\cdot 61^{5} + 47\cdot 61^{6} + 34\cdot 61^{7} + 47\cdot 61^{8} + 53\cdot 61^{9} + 53\cdot 61^{10} + 13\cdot 61^{11} + 2\cdot 61^{12} +O\left(61^{ 13 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 4 a + 59 + \left(29 a + 17\right)\cdot 61 + \left(60 a + 45\right)\cdot 61^{2} + \left(4 a + 27\right)\cdot 61^{3} + \left(41 a + 12\right)\cdot 61^{4} + \left(35 a + 33\right)\cdot 61^{5} + \left(49 a + 23\right)\cdot 61^{6} + \left(4 a + 22\right)\cdot 61^{7} + \left(37 a + 14\right)\cdot 61^{8} + \left(32 a + 2\right)\cdot 61^{9} + \left(a + 46\right)\cdot 61^{10} + \left(47 a + 7\right)\cdot 61^{11} + \left(56 a + 56\right)\cdot 61^{12} +O\left(61^{ 13 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 35 a + 13 + \left(17 a + 39\right)\cdot 61 + \left(35 a + 21\right)\cdot 61^{2} + \left(47 a + 24\right)\cdot 61^{3} + \left(11 a + 48\right)\cdot 61^{4} + \left(42 a + 45\right)\cdot 61^{5} + \left(9 a + 46\right)\cdot 61^{6} + \left(56 a + 37\right)\cdot 61^{7} + \left(26 a + 14\right)\cdot 61^{8} + \left(9 a + 39\right)\cdot 61^{9} + \left(11 a + 29\right)\cdot 61^{10} + \left(11 a + 30\right)\cdot 61^{11} + \left(26 a + 53\right)\cdot 61^{12} +O\left(61^{ 13 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 42 + 33\cdot 61 + 40\cdot 61^{2} + 29\cdot 61^{3} + 15\cdot 61^{4} + 55\cdot 61^{5} + 54\cdot 61^{6} + 43\cdot 61^{7} + 23\cdot 61^{8} + 28\cdot 61^{9} + 21\cdot 61^{10} + 24\cdot 61^{11} + 60\cdot 61^{12} +O\left(61^{ 13 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,7,8,5,6,3,4)$ |
| $(2,6)(4,8)$ |
| $(3,7)(4,8)$ |
| $(1,5)(2,6)(3,7)(4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $-4$ |
| $2$ | $2$ | $(2,6)(4,8)$ | $0$ |
| $4$ | $2$ | $(3,7)(4,8)$ | $0$ |
| $4$ | $2$ | $(1,7)(2,8)(3,5)(4,6)$ | $0$ |
| $2$ | $4$ | $(1,7,5,3)(2,8,6,4)$ | $0$ |
| $2$ | $4$ | $(1,7,5,3)(2,4,6,8)$ | $0$ |
| $4$ | $8$ | $(1,2,7,8,5,6,3,4)$ | $0$ |
| $4$ | $8$ | $(1,8,3,2,5,4,7,6)$ | $0$ |
| $4$ | $8$ | $(1,2,7,4,5,6,3,8)$ | $0$ |
| $4$ | $8$ | $(1,4,3,2,5,8,7,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
