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:
| $r_{ 1 }$ |
$=$ |
$ 4 + 17\cdot 61 + 26\cdot 61^{2} + 49\cdot 61^{3} + 38\cdot 61^{4} + 51\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 48 a + 45 + \left(43 a + 5\right)\cdot 61 + \left(37 a + 49\right)\cdot 61^{2} + \left(4 a + 37\right)\cdot 61^{3} + \left(40 a + 13\right)\cdot 61^{4} + \left(17 a + 57\right)\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 13 a + 32 + \left(17 a + 1\right)\cdot 61 + \left(23 a + 43\right)\cdot 61^{2} + \left(56 a + 4\right)\cdot 61^{3} + \left(20 a + 49\right)\cdot 61^{4} + \left(43 a + 34\right)\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 35 a + 53 + \left(39 a + 43\right)\cdot 61 + \left(19 a + 26\right)\cdot 61^{2} + \left(42 a + 21\right)\cdot 61^{3} + \left(59 a + 16\right)\cdot 61^{4} + \left(26 a + 55\right)\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 26 a + 27 + \left(21 a + 48\right)\cdot 61 + \left(41 a + 6\right)\cdot 61^{2} + \left(18 a + 44\right)\cdot 61^{3} + \left(a + 33\right)\cdot 61^{4} + \left(34 a + 22\right)\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 21 a + 17 + \left(25 a + 5\right)\cdot 61 + \left(13 a + 57\right)\cdot 61^{2} + \left(9 a + 42\right)\cdot 61^{3} + \left(55 a + 19\right)\cdot 61^{4} + \left(22 a + 11\right)\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 29 + 51\cdot 61 + 50\cdot 61^{2} + 4\cdot 61^{3} + 7\cdot 61^{4} + 32\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 40 a + 38 + \left(35 a + 9\right)\cdot 61 + \left(47 a + 45\right)\cdot 61^{2} + \left(51 a + 38\right)\cdot 61^{3} + \left(5 a + 4\right)\cdot 61^{4} + \left(38 a + 40\right)\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,7)(2,6)(5,8)$ |
| $(1,3,7,4)(2,8,5,6)$ |
| $(1,5,7,2)(3,8,4,6)$ |
| $(1,7)(2,5)(3,4)(6,8)$ |
| $(1,2,8)(5,6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,5)(3,4)(6,8)$ | $-2$ |
| $12$ | $2$ | $(1,7)(2,6)(5,8)$ | $0$ |
| $8$ | $3$ | $(1,5,4)(2,3,7)$ | $-1$ |
| $6$ | $4$ | $(1,3,7,4)(2,8,5,6)$ | $0$ |
| $8$ | $6$ | $(1,3,5,7,4,2)(6,8)$ | $1$ |
| $6$ | $8$ | $(1,2,3,8,7,5,4,6)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
| $6$ | $8$ | $(1,5,3,6,7,2,4,8)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
