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^{2} + 63 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 54 a + 31 + \left(61 a + 56\right)\cdot 67 + \left(66 a + 2\right)\cdot 67^{2} + \left(58 a + 53\right)\cdot 67^{3} + \left(56 a + 34\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 + 10\cdot 67 + 51\cdot 67^{2} + 15\cdot 67^{3} + 66\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 22 a + 51 + \left(5 a + 62\right)\cdot 67 + \left(11 a + 45\right)\cdot 67^{2} + \left(42 a + 49\right)\cdot 67^{3} + 7 a\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 5 a + 21 + \left(3 a + 10\right)\cdot 67 + \left(37 a + 32\right)\cdot 67^{2} + \left(42 a + 61\right)\cdot 67^{3} + \left(56 a + 28\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 45 a + 5 + \left(61 a + 62\right)\cdot 67 + \left(55 a + 17\right)\cdot 67^{2} + \left(24 a + 6\right)\cdot 67^{3} + \left(59 a + 56\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 62 a + 41 + \left(63 a + 17\right)\cdot 67 + \left(29 a + 43\right)\cdot 67^{2} + \left(24 a + 60\right)\cdot 67^{3} + \left(10 a + 11\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 13 a + 46 + \left(5 a + 48\right)\cdot 67 + 7\cdot 67^{2} + \left(8 a + 21\right)\cdot 67^{3} + \left(10 a + 2\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 7 }$
| Cycle notation |
| $(1,2,3,4,5,6,7)$ |
| $(1,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 7 }$
| Character value |
| $1$ | $1$ | $()$ | $21$ |
| $21$ | $2$ | $(1,2)$ | $-1$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $3$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $70$ | $3$ | $(1,2,3)$ | $-3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $210$ | $4$ | $(1,2,3,4)$ | $1$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $-1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
