Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$: $ x^{2} + 60 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 12 + 20\cdot 61 + 18\cdot 61^{2} + 40\cdot 61^{3} + 35\cdot 61^{4} + 30\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 21 a + 45 + \left(29 a + 46\right)\cdot 61 + \left(28 a + 21\right)\cdot 61^{2} + \left(55 a + 27\right)\cdot 61^{3} + \left(28 a + 56\right)\cdot 61^{4} + \left(29 a + 14\right)\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 49 + 20\cdot 61 + 9\cdot 61^{2} + 54\cdot 61^{3} + 3\cdot 61^{4} + 8\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 40 a + 5 + \left(31 a + 55\right)\cdot 61 + \left(32 a + 20\right)\cdot 61^{2} + \left(5 a + 54\right)\cdot 61^{3} + \left(32 a + 29\right)\cdot 61^{4} + \left(31 a + 15\right)\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 13 a + \left(13 a + 20\right)\cdot 61 + \left(5 a + 60\right)\cdot 61^{2} + \left(39 a + 16\right)\cdot 61^{3} + \left(58 a + 49\right)\cdot 61^{4} + \left(49 a + 30\right)\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 48 a + 13 + \left(47 a + 20\right)\cdot 61 + \left(55 a + 52\right)\cdot 61^{2} + \left(21 a + 50\right)\cdot 61^{3} + \left(2 a + 7\right)\cdot 61^{4} + \left(11 a + 22\right)\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)(3,6)$ |
| $(3,5,6)$ |
| $(1,3,2,6,4,5)$ |
| $(1,2)(3,5)$ |
| $(2,4)(3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $3$ | $2$ | $(1,6)(2,5)(3,4)$ | $0$ |
| $3$ | $2$ | $(1,5)(2,6)(3,4)$ | $0$ |
| $9$ | $2$ | $(1,4)(3,6)$ | $0$ |
| $2$ | $3$ | $(1,2,4)(3,6,5)$ | $-2$ |
| $2$ | $3$ | $(1,2,4)(3,5,6)$ | $-2$ |
| $4$ | $3$ | $(1,2,4)$ | $1$ |
| $6$ | $6$ | $(1,3,2,6,4,5)$ | $0$ |
| $6$ | $6$ | $(1,6,2,3,4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
