Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$: $ x^{3} + 6 x + 65 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 33 a^{2} + 38 + \left(2 a^{2} + 22 a + 36\right)\cdot 67 + \left(27 a^{2} + a + 63\right)\cdot 67^{2} + \left(54 a^{2} + 6 a + 59\right)\cdot 67^{3} + \left(6 a + 36\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 48 a^{2} + 28 a + 37 + \left(a^{2} + 59 a + 1\right)\cdot 67 + \left(64 a^{2} + 21 a + 13\right)\cdot 67^{2} + \left(31 a^{2} + 5 a + 46\right)\cdot 67^{3} + \left(48 a^{2} + 28 a + 7\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 9 a^{2} + 56 a + 15 + \left(27 a^{2} + 51 a + 36\right)\cdot 67 + \left(15 a^{2} + 17 a + 19\right)\cdot 67^{2} + \left(57 a^{2} + 27 a + 13\right)\cdot 67^{3} + \left(23 a^{2} + 32 a + 43\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 61 a^{2} + 25 a + 16 + \left(a^{2} + 26 a + 34\right)\cdot 67 + \left(28 a^{2} + 63 a\right)\cdot 67^{2} + \left(40 a^{2} + 26 a + 4\right)\cdot 67^{3} + \left(39 a^{2} + 15 a + 58\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 40 a^{2} + 42 a + 66 + \left(62 a^{2} + 18 a + 8\right)\cdot 67 + \left(11 a^{2} + 2 a + 3\right)\cdot 67^{2} + \left(39 a^{2} + 34 a + 66\right)\cdot 67^{3} + \left(26 a^{2} + 45 a + 5\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 10 a^{2} + 50 a + 19 + \left(38 a^{2} + 22 a + 13\right)\cdot 67 + \left(54 a^{2} + 27 a + 42\right)\cdot 67^{2} + \left(44 a^{2} + 34 a + 30\right)\cdot 67^{3} + \left(61 a^{2} + 6 a + 60\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 3 a^{2} + 50 a + 39 + \left(20 a^{2} + 64 a + 36\right)\cdot 67 + \left(35 a^{2} + 66 a + 4\right)\cdot 67^{2} + \left(8 a^{2} + 38 a + 50\right)\cdot 67^{3} + \left(39 a^{2} + 37 a + 18\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 5 a^{2} + 13 a + 47 + \left(9 a^{2} + 39 a + 59\right)\cdot 67 + \left(45 a^{2} + 11 a + 43\right)\cdot 67^{2} + \left(64 a^{2} + 37 a + 6\right)\cdot 67^{3} + \left(43 a^{2} + 64 a + 38\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 59 a^{2} + 4 a + 62 + \left(37 a^{2} + 30 a + 40\right)\cdot 67 + \left(53 a^{2} + 55 a + 10\right)\cdot 67^{2} + \left(60 a^{2} + 57 a + 58\right)\cdot 67^{3} + \left(50 a^{2} + 31 a + 65\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,7,3,4,9,2,5,8,6)$ |
| $(1,4,5)(2,6,3)(7,9,8)$ |
| $(1,6)(2,4)(3,5)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $9$ | $2$ | $(1,6)(2,4)(3,5)(7,8)$ | $0$ |
| $2$ | $3$ | $(1,4,5)(2,6,3)(7,9,8)$ | $-1$ |
| $2$ | $9$ | $(1,7,3,4,9,2,5,8,6)$ | $\zeta_{9}^{5} + \zeta_{9}^{4}$ |
| $2$ | $9$ | $(1,3,9,5,6,7,4,2,8)$ | $-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
| $2$ | $9$ | $(1,9,6,4,8,3,5,7,2)$ | $-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
The blue line marks the conjugacy class containing complex conjugation.
