Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 71 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 71 }$: $ x^{2} + 69 x + 7 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 59 a + 55 + \left(13 a + 60\right)\cdot 71 + \left(44 a + 69\right)\cdot 71^{2} + \left(12 a + 5\right)\cdot 71^{3} + \left(69 a + 26\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 56 + 51\cdot 71 + 69\cdot 71^{2} + 6\cdot 71^{3} + 35\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 21 + 64\cdot 71 + 26\cdot 71^{2} + 13\cdot 71^{3} + 22\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 12 a + 31 + \left(57 a + 29\right)\cdot 71 + \left(26 a + 2\right)\cdot 71^{2} + \left(58 a + 58\right)\cdot 71^{3} + \left(a + 9\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 61 a + 35 + \left(48 a + 20\right)\cdot 71 + \left(56 a + 25\right)\cdot 71^{2} + \left(48 a + 8\right)\cdot 71^{3} + \left(26 a + 22\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 10 a + 15 + \left(22 a + 57\right)\cdot 71 + \left(14 a + 18\right)\cdot 71^{2} + \left(22 a + 49\right)\cdot 71^{3} + \left(44 a + 26\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,4)$ |
| $(1,3)(2,5)(4,6)$ |
| $(1,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,3)(2,5)(4,6)$ | $0$ |
| $6$ | $2$ | $(2,4)$ | $-2$ |
| $9$ | $2$ | $(2,4)(5,6)$ | $0$ |
| $4$ | $3$ | $(1,2,4)(3,5,6)$ | $-2$ |
| $4$ | $3$ | $(3,5,6)$ | $1$ |
| $18$ | $4$ | $(1,3)(2,6,4,5)$ | $0$ |
| $12$ | $6$ | $(1,3,2,5,4,6)$ | $0$ |
| $12$ | $6$ | $(2,4)(3,5,6)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
