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:
| $r_{ 1 }$ |
$=$ |
$ 61 a + 18 + \left(66 a + 45\right)\cdot 67 + \left(59 a + 13\right)\cdot 67^{2} + \left(41 a + 16\right)\cdot 67^{3} + \left(20 a + 17\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 58 a + 43 + \left(2 a + 8\right)\cdot 67 + \left(33 a + 26\right)\cdot 67^{2} + \left(66 a + 54\right)\cdot 67^{3} + \left(59 a + 56\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 6 a + 61 + 50\cdot 67 + \left(7 a + 52\right)\cdot 67^{2} + \left(25 a + 56\right)\cdot 67^{3} + \left(46 a + 57\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 6 + 20\cdot 67^{2} + 54\cdot 67^{3} + 39\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 9 a + 7 + \left(64 a + 29\right)\cdot 67 + \left(33 a + 21\right)\cdot 67^{2} + 19\cdot 67^{3} + \left(7 a + 29\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $10$ |
$2$ |
$(1,2)$ |
$2$ |
| $15$ |
$2$ |
$(1,2)(3,4)$ |
$0$ |
| $20$ |
$3$ |
$(1,2,3)$ |
$1$ |
| $30$ |
$4$ |
$(1,2,3,4)$ |
$0$ |
| $24$ |
$5$ |
$(1,2,3,4,5)$ |
$-1$ |
| $20$ |
$6$ |
$(1,2,3)(4,5)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
