Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 71 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 71 }$: $ x^{2} + 69 x + 7 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 14\cdot 71 + 20\cdot 71^{2} + 50\cdot 71^{3} + 67\cdot 71^{4} + 50\cdot 71^{5} + 65\cdot 71^{6} +O\left(71^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 28 a + 44 + \left(50 a + 41\right)\cdot 71 + \left(58 a + 47\right)\cdot 71^{2} + \left(47 a + 6\right)\cdot 71^{3} + \left(38 a + 19\right)\cdot 71^{4} + \left(30 a + 14\right)\cdot 71^{5} + \left(70 a + 13\right)\cdot 71^{6} +O\left(71^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 43 a + 29 + \left(20 a + 43\right)\cdot 71 + \left(12 a + 43\right)\cdot 71^{2} + \left(23 a + 43\right)\cdot 71^{3} + \left(32 a + 48\right)\cdot 71^{4} + \left(40 a + 36\right)\cdot 71^{5} + 52\cdot 71^{6} +O\left(71^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 69 + 56\cdot 71 + 50\cdot 71^{2} + 20\cdot 71^{3} + 3\cdot 71^{4} + 20\cdot 71^{5} + 5\cdot 71^{6} +O\left(71^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 43 a + 27 + \left(20 a + 29\right)\cdot 71 + \left(12 a + 23\right)\cdot 71^{2} + \left(23 a + 64\right)\cdot 71^{3} + \left(32 a + 51\right)\cdot 71^{4} + \left(40 a + 56\right)\cdot 71^{5} + 57\cdot 71^{6} +O\left(71^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 28 a + 42 + \left(50 a + 27\right)\cdot 71 + \left(58 a + 27\right)\cdot 71^{2} + \left(47 a + 27\right)\cdot 71^{3} + \left(38 a + 22\right)\cdot 71^{4} + \left(30 a + 34\right)\cdot 71^{5} + \left(70 a + 18\right)\cdot 71^{6} +O\left(71^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,3)(5,6)$ |
| $(1,2)(3,6)(4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-2$ |
| $3$ | $2$ | $(1,2)(3,6)(4,5)$ | $0$ |
| $3$ | $2$ | $(1,6)(3,4)$ | $0$ |
| $2$ | $3$ | $(1,5,6)(2,3,4)$ | $-1$ |
| $2$ | $6$ | $(1,3,5,4,6,2)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
