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:
| $r_{ 1 }$ |
$=$ |
$ 31 a^{2} + 5 a + 14 + \left(8 a^{2} + 48 a + 49\right)\cdot 61 + \left(15 a^{2} + 21 a + 22\right)\cdot 61^{2} + \left(27 a^{2} + 13\right)\cdot 61^{3} + \left(11 a^{2} + 9 a + 44\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 55 a^{2} + 59 a + 4 + \left(32 a^{2} + 44 a + 41\right)\cdot 61 + \left(6 a^{2} + 23 a + 43\right)\cdot 61^{2} + \left(a^{2} + 49 a + 33\right)\cdot 61^{3} + \left(50 a^{2} + 59 a\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 3 a^{2} + 4 a + 33 + \left(37 a^{2} + 32 a + 16\right)\cdot 61 + \left(60 a^{2} + 16 a + 58\right)\cdot 61^{2} + \left(40 a^{2} + 34 a + 12\right)\cdot 61^{3} + \left(3 a^{2} + 40 a + 42\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 36 a^{2} + 58 a + 17 + \left(19 a^{2} + 28 a + 60\right)\cdot 61 + \left(39 a^{2} + 15 a + 33\right)\cdot 61^{2} + \left(32 a^{2} + 11 a + 38\right)\cdot 61^{3} + \left(60 a^{2} + 53 a + 49\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 31 a^{2} + 43 a + 53 + \left(4 a^{2} + 26 a + 45\right)\cdot 61 + \left(31 a^{2} + 25 a + 10\right)\cdot 61^{2} + \left(20 a^{2} + 13 a + 42\right)\cdot 61^{3} + \left(52 a^{2} + 52 a + 4\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 23 a^{2} + 35 a + 36 + \left(3 a^{2} + 16 a + 40\right)\cdot 61 + \left(40 a^{2} + 3 a + 52\right)\cdot 61^{2} + \left(21 a^{2} + 54 a + 6\right)\cdot 61^{3} + \left(44 a^{2} + 45 a + 8\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 7 a^{2} + 44 a + 2 + \left(53 a^{2} + 17 a + 49\right)\cdot 61 + \left(50 a^{2} + 32 a + 21\right)\cdot 61^{2} + \left(18 a^{2} + 54 a + 34\right)\cdot 61^{3} + \left(25 a^{2} + 23 a\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 40 a^{2} + 59 a + 43 + \left(40 a^{2} + 11 a + 53\right)\cdot 61 + \left(42 a^{2} + 27 a + 55\right)\cdot 61^{2} + \left(50 a^{2} + 51 a + 37\right)\cdot 61^{3} + \left(9 a^{2} + 41 a + 50\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 18 a^{2} + 59 a + 42 + \left(44 a^{2} + 16 a + 9\right)\cdot 61 + \left(18 a^{2} + 17 a + 5\right)\cdot 61^{2} + \left(30 a^{2} + 36 a + 24\right)\cdot 61^{3} + \left(47 a^{2} + 39 a + 43\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,4,2)(3,8,9)(5,7,6)$ |
| $(1,2)(3,6)(5,9)(7,8)$ |
| $(1,7,3,4,6,8,2,5,9)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $9$ | $2$ | $(1,2)(3,6)(5,9)(7,8)$ | $0$ |
| $2$ | $3$ | $(1,4,2)(3,8,9)(5,7,6)$ | $-1$ |
| $2$ | $9$ | $(1,7,3,4,6,8,2,5,9)$ | $\zeta_{9}^{5} + \zeta_{9}^{4}$ |
| $2$ | $9$ | $(1,3,6,2,9,7,4,8,5)$ | $-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
| $2$ | $9$ | $(1,6,9,4,5,3,2,7,8)$ | $-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
The blue line marks the conjugacy class containing complex conjugation.
