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:
| $r_{ 1 }$ |
$=$ |
$ 29 a + 26 + \left(28 a + 9\right)\cdot 61 + \left(43 a + 40\right)\cdot 61^{2} + \left(49 a + 44\right)\cdot 61^{3} + \left(10 a + 23\right)\cdot 61^{4} + \left(52 a + 59\right)\cdot 61^{5} + \left(3 a + 44\right)\cdot 61^{6} + \left(39 a + 60\right)\cdot 61^{7} +O\left(61^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 53 + 4\cdot 61 + 56\cdot 61^{2} + 20\cdot 61^{3} + 46\cdot 61^{4} + 36\cdot 61^{5} + 9\cdot 61^{6} + 8\cdot 61^{7} +O\left(61^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 22 a + 24 + \left(44 a + 47\right)\cdot 61 + \left(13 a + 17\right)\cdot 61^{2} + \left(60 a + 27\right)\cdot 61^{3} + \left(5 a + 34\right)\cdot 61^{4} + \left(58 a + 16\right)\cdot 61^{5} + \left(46 a + 31\right)\cdot 61^{6} + 49\cdot 61^{7} +O\left(61^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 32 a + 55 + \left(32 a + 8\right)\cdot 61 + \left(17 a + 55\right)\cdot 61^{2} + \left(11 a + 50\right)\cdot 61^{3} + \left(50 a + 45\right)\cdot 61^{4} + \left(8 a + 39\right)\cdot 61^{5} + \left(57 a + 57\right)\cdot 61^{6} + \left(21 a + 34\right)\cdot 61^{7} +O\left(61^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 42 + 42\cdot 61 + 26\cdot 61^{2} + 26\cdot 61^{3} + 52\cdot 61^{4} + 22\cdot 61^{5} + 19\cdot 61^{6} + 26\cdot 61^{7} +O\left(61^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 39 a + 46 + \left(16 a + 8\right)\cdot 61 + \left(47 a + 48\right)\cdot 61^{2} + 12\cdot 61^{3} + \left(55 a + 41\right)\cdot 61^{4} + \left(2 a + 7\right)\cdot 61^{5} + \left(14 a + 20\right)\cdot 61^{6} + \left(60 a + 3\right)\cdot 61^{7} +O\left(61^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,3)(4,5)$ |
| $(1,2)(3,4)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,6)(2,5)(3,4)$ |
$-2$ |
| $3$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$0$ |
| $3$ |
$2$ |
$(1,4)(3,6)$ |
$0$ |
| $2$ |
$3$ |
$(1,5,4)(2,3,6)$ |
$-1$ |
| $2$ |
$6$ |
$(1,3,5,6,4,2)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
