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:
| $r_{ 1 }$ |
$=$ |
$ 8 a + 44 + \left(13 a + 4\right)\cdot 71 + \left(36 a + 37\right)\cdot 71^{2} + \left(69 a + 11\right)\cdot 71^{3} + 7\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 64 a + 50 + 47\cdot 71 + \left(2 a + 8\right)\cdot 71^{2} + \left(33 a + 40\right)\cdot 71^{3} + \left(55 a + 54\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 63 a + 60 + \left(57 a + 22\right)\cdot 71 + \left(34 a + 25\right)\cdot 71^{2} + \left(a + 43\right)\cdot 71^{3} + \left(70 a + 10\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 7 a + 36 + \left(70 a + 56\right)\cdot 71 + \left(68 a + 11\right)\cdot 71^{2} + \left(37 a + 33\right)\cdot 71^{3} + \left(15 a + 61\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 57 + 37\cdot 71 + 50\cdot 71^{2} + 68\cdot 71^{3} + 25\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 39 + 43\cdot 71 + 8\cdot 71^{2} + 16\cdot 71^{3} + 53\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,4)$ |
| $(1,2)(3,4)(5,6)$ |
| $(2,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $6$ | $2$ | $(3,6)$ | $-2$ |
| $9$ | $2$ | $(3,6)(4,5)$ | $0$ |
| $4$ | $3$ | $(1,3,6)(2,4,5)$ | $-2$ |
| $4$ | $3$ | $(1,3,6)$ | $1$ |
| $18$ | $4$ | $(1,2)(3,5,6,4)$ | $0$ |
| $12$ | $6$ | $(1,4,3,5,6,2)$ | $0$ |
| $12$ | $6$ | $(2,4,5)(3,6)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
