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:
| $r_{ 1 }$ |
$=$ |
$ 9 + 40\cdot 61 + 19\cdot 61^{3} + 56\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 55 a + 5 + 11\cdot 61 + \left(31 a + 18\right)\cdot 61^{2} + \left(34 a + 10\right)\cdot 61^{3} + \left(37 a + 34\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 10 + 59\cdot 61 + 14\cdot 61^{2} + 3\cdot 61^{3} + 24\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 57 a + 52 + \left(14 a + 17\right)\cdot 61 + \left(52 a + 1\right)\cdot 61^{2} + \left(25 a + 51\right)\cdot 61^{3} + \left(21 a + 17\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 6 a + 60 + \left(60 a + 17\right)\cdot 61 + \left(29 a + 48\right)\cdot 61^{2} + \left(26 a + 13\right)\cdot 61^{3} + \left(23 a + 37\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 4 a + 48 + \left(46 a + 36\right)\cdot 61 + \left(8 a + 38\right)\cdot 61^{2} + \left(35 a + 24\right)\cdot 61^{3} + \left(39 a + 13\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)$ |
| $(2,3)$ |
| $(2,3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $-2$ |
| $6$ | $2$ | $(3,5)$ | $0$ |
| $9$ | $2$ | $(3,5)(4,6)$ | $0$ |
| $4$ | $3$ | $(1,4,6)(2,3,5)$ | $1$ |
| $4$ | $3$ | $(1,4,6)$ | $-2$ |
| $18$ | $4$ | $(1,2)(3,6,5,4)$ | $0$ |
| $12$ | $6$ | $(1,3,4,5,6,2)$ | $1$ |
| $12$ | $6$ | $(1,4,6)(3,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
