Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$: $ x^{2} + 63 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 25 a + 52 + \left(8 a + 29\right)\cdot 67 + \left(29 a + 44\right)\cdot 67^{2} + \left(40 a + 47\right)\cdot 67^{3} + \left(51 a + 15\right)\cdot 67^{4} + \left(10 a + 15\right)\cdot 67^{5} + \left(4 a + 53\right)\cdot 67^{6} + \left(36 a + 8\right)\cdot 67^{7} +O\left(67^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 33 + 17\cdot 67 + 21\cdot 67^{2} + 26\cdot 67^{3} + 39\cdot 67^{4} + 35\cdot 67^{5} + 46\cdot 67^{6} + 5\cdot 67^{7} +O\left(67^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 20 a + 62 + \left(50 a + 18\right)\cdot 67 + \left(6 a + 5\right)\cdot 67^{2} + \left(28 a + 29\right)\cdot 67^{3} + \left(4 a + 10\right)\cdot 67^{4} + \left(14 a + 62\right)\cdot 67^{5} + 28\cdot 67^{6} + \left(25 a + 15\right)\cdot 67^{7} +O\left(67^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 29 + 30\cdot 67 + 62\cdot 67^{2} + 50\cdot 67^{3} + 20\cdot 67^{4} + 34\cdot 67^{5} + 64\cdot 67^{6} + 40\cdot 67^{7} +O\left(67^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 42 a + 18 + \left(58 a + 38\right)\cdot 67 + \left(37 a + 18\right)\cdot 67^{2} + \left(26 a + 46\right)\cdot 67^{3} + \left(15 a + 47\right)\cdot 67^{4} + \left(56 a + 6\right)\cdot 67^{5} + \left(62 a + 59\right)\cdot 67^{6} + \left(30 a + 14\right)\cdot 67^{7} +O\left(67^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 47 a + 8 + \left(16 a + 66\right)\cdot 67 + \left(60 a + 48\right)\cdot 67^{2} + 38 a\cdot 67^{3} + 62 a\cdot 67^{4} + \left(52 a + 47\right)\cdot 67^{5} + \left(66 a + 15\right)\cdot 67^{6} + \left(41 a + 48\right)\cdot 67^{7} +O\left(67^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,4)$ |
| $(2,3)(4,5)$ |
| $(1,2,3)(4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,6)(2,4)(3,5)$ | $-3$ |
| $3$ | $2$ | $(2,4)(3,5)$ | $-1$ |
| $3$ | $2$ | $(3,5)$ | $1$ |
| $6$ | $2$ | $(2,3)(4,5)$ | $1$ |
| $6$ | $2$ | $(1,2)(3,5)(4,6)$ | $-1$ |
| $8$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $6$ | $4$ | $(2,3,4,5)$ | $1$ |
| $6$ | $4$ | $(1,6)(2,3,4,5)$ | $-1$ |
| $8$ | $6$ | $(1,3,4,6,5,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
