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^{3} + 7 x + 59 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 56 a^{2} + 3 a + 44 + \left(29 a^{2} + 49 a + 10\right)\cdot 61 + \left(57 a^{2} + 29 a + 32\right)\cdot 61^{2} + \left(43 a^{2} + 36 a + 52\right)\cdot 61^{3} + \left(49 a^{2} + 16 a + 42\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 35 a^{2} + 26 a + 3 + \left(10 a^{2} + 5 a + 36\right)\cdot 61 + \left(28 a^{2} + 30 a + 26\right)\cdot 61^{2} + \left(4 a^{2} + 57 a + 10\right)\cdot 61^{3} + \left(8 a + 14\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 41 a^{2} + 49 a + 35 + \left(19 a^{2} + a + 3\right)\cdot 61 + \left(57 a^{2} + 11 a + 11\right)\cdot 61^{2} + \left(47 a^{2} + 38 a + 10\right)\cdot 61^{3} + \left(6 a^{2} + 4 a + 5\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 25 a^{2} + 9 a + 1 + \left(11 a^{2} + 10 a + 26\right)\cdot 61 + \left(7 a^{2} + 20 a + 41\right)\cdot 61^{2} + \left(30 a^{2} + 47 a + 28\right)\cdot 61^{3} + \left(4 a^{2} + 39 a + 55\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 11 a^{2} + 30 a + 23 + \left(8 a^{2} + 31 a + 58\right)\cdot 61 + \left(52 a^{2} + 17 a + 14\right)\cdot 61^{2} + \left(53 a^{2} + 19 a + 48\right)\cdot 61^{3} + \left(24 a^{2} + 9 a + 6\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 44 a^{2} + 2 a + 55 + \left(35 a^{2} + 7 a + 3\right)\cdot 61 + \left(55 a^{2} + 16 a + 31\right)\cdot 61^{2} + \left(12 a^{2} + 35 a + 60\right)\cdot 61^{3} + \left(42 a^{2} + 56 a + 46\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 23 a^{2} + 55 a + 8 + \left(35 a^{2} + a + 50\right)\cdot 61 + \left(42 a^{2} + 34 a + 32\right)\cdot 61^{2} + \left(14 a^{2} + 19 a + 17\right)\cdot 61^{3} + \left(18 a^{2} + 29 a + 58\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 6 a^{2} + 29 a + 20 + \left(17 a^{2} + 22 a + 59\right)\cdot 61 + \left(14 a^{2} + 27 a\right)\cdot 61^{2} + \left(55 a^{2} + 6 a + 14\right)\cdot 61^{3} + \left(54 a^{2} + 56 a + 45\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 3 a^{2} + 41 a + 57 + \left(15 a^{2} + 53 a + 56\right)\cdot 61 + \left(51 a^{2} + 57 a + 52\right)\cdot 61^{2} + \left(41 a^{2} + 44 a + 1\right)\cdot 61^{3} + \left(42 a^{2} + 22 a + 30\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,2)(3,7)(4,9)(5,6)$ |
| $(1,2,5,4,7,8,3,9,6)$ |
| $(1,4,3)(2,7,9)(5,8,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $9$ | $2$ | $(1,2)(3,7)(4,9)(5,6)$ | $0$ |
| $2$ | $3$ | $(1,4,3)(2,7,9)(5,8,6)$ | $-1$ |
| $2$ | $9$ | $(1,2,5,4,7,8,3,9,6)$ | $\zeta_{9}^{5} + \zeta_{9}^{4}$ |
| $2$ | $9$ | $(1,5,7,3,6,2,4,8,9)$ | $-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
| $2$ | $9$ | $(1,7,6,4,9,5,3,2,8)$ | $-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
The blue line marks the conjugacy class containing complex conjugation.
