Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 14.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$: $ x^{2} + 60 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 22 + 32\cdot 61 + 8\cdot 61^{2} + 45\cdot 61^{3} + 20\cdot 61^{4} + 3\cdot 61^{5} + 15\cdot 61^{6} + 44\cdot 61^{7} + 42\cdot 61^{8} + 39\cdot 61^{9} + 30\cdot 61^{10} + 27\cdot 61^{11} + 4\cdot 61^{12} + 49\cdot 61^{13} +O\left(61^{ 14 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 47 a + 5 + \left(25 a + 38\right)\cdot 61 + \left(44 a + 38\right)\cdot 61^{2} + \left(54 a + 11\right)\cdot 61^{3} + \left(2 a + 15\right)\cdot 61^{4} + \left(31 a + 46\right)\cdot 61^{5} + \left(60 a + 35\right)\cdot 61^{6} + \left(46 a + 37\right)\cdot 61^{7} + \left(8 a + 32\right)\cdot 61^{8} + \left(34 a + 25\right)\cdot 61^{9} + \left(57 a + 6\right)\cdot 61^{10} + \left(26 a + 6\right)\cdot 61^{11} + \left(19 a + 31\right)\cdot 61^{12} + \left(a + 42\right)\cdot 61^{13} +O\left(61^{ 14 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 14 a + 52 + \left(35 a + 16\right)\cdot 61 + \left(16 a + 57\right)\cdot 61^{2} + \left(6 a + 21\right)\cdot 61^{3} + \left(58 a + 24\right)\cdot 61^{4} + \left(29 a + 13\right)\cdot 61^{5} + 4\cdot 61^{6} + \left(14 a + 24\right)\cdot 61^{7} + \left(52 a + 55\right)\cdot 61^{8} + \left(26 a + 50\right)\cdot 61^{9} + \left(3 a + 29\right)\cdot 61^{10} + \left(34 a + 36\right)\cdot 61^{11} + \left(41 a + 23\right)\cdot 61^{12} + \left(59 a + 24\right)\cdot 61^{13} +O\left(61^{ 14 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 50 + 12\cdot 61 + 26\cdot 61^{2} + 50\cdot 61^{3} + 59\cdot 61^{4} + 6\cdot 61^{5} + 50\cdot 61^{6} + 49\cdot 61^{7} + 52\cdot 61^{8} + 11\cdot 61^{9} + 17\cdot 61^{10} + 18\cdot 61^{11} + 28\cdot 61^{12} + 5\cdot 61^{13} +O\left(61^{ 14 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 11 a + 22 + \left(6 a + 13\right)\cdot 61 + \left(41 a + 39\right)\cdot 61^{2} + 16\cdot 61^{3} + \left(36 a + 44\right)\cdot 61^{4} + \left(2 a + 42\right)\cdot 61^{5} + \left(55 a + 12\right)\cdot 61^{6} + \left(47 a + 17\right)\cdot 61^{7} + \left(9 a + 49\right)\cdot 61^{8} + \left(42 a + 41\right)\cdot 61^{9} + \left(40 a + 19\right)\cdot 61^{10} + \left(4 a + 4\right)\cdot 61^{11} + \left(7 a + 16\right)\cdot 61^{12} + \left(30 a + 19\right)\cdot 61^{13} +O\left(61^{ 14 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 50 a + 33 + \left(54 a + 8\right)\cdot 61 + \left(19 a + 13\right)\cdot 61^{2} + \left(60 a + 37\right)\cdot 61^{3} + \left(24 a + 18\right)\cdot 61^{4} + \left(58 a + 9\right)\cdot 61^{5} + \left(5 a + 4\right)\cdot 61^{6} + \left(13 a + 10\right)\cdot 61^{7} + \left(51 a + 11\right)\cdot 61^{8} + \left(18 a + 13\right)\cdot 61^{9} + \left(20 a + 18\right)\cdot 61^{10} + \left(56 a + 29\right)\cdot 61^{11} + \left(53 a + 18\right)\cdot 61^{12} + \left(30 a + 42\right)\cdot 61^{13} +O\left(61^{ 14 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(4,5,6)$ |
| $(1,2,3)$ |
| $(1,5,2,4)(3,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $9$ | $2$ | $(1,2)(4,5)$ | $0$ |
| $4$ | $3$ | $(1,2,3)$ | $1$ |
| $4$ | $3$ | $(1,2,3)(4,5,6)$ | $-2$ |
| $9$ | $4$ | $(1,5,2,4)(3,6)$ | $0$ |
| $9$ | $4$ | $(1,4,2,5)(3,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
