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 + 5 + \left(57 a + 36\right)\cdot 71 + \left(22 a + 7\right)\cdot 71^{2} + \left(5 a + 29\right)\cdot 71^{3} + \left(40 a + 28\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 26 a + 19 + \left(56 a + 57\right)\cdot 71 + \left(28 a + 60\right)\cdot 71^{2} + \left(31 a + 61\right)\cdot 71^{3} + \left(29 a + 44\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 45 a + \left(14 a + 2\right)\cdot 71 + \left(42 a + 62\right)\cdot 71^{2} + \left(39 a + 24\right)\cdot 71^{3} + \left(41 a + 1\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 3 a + 66 + \left(22 a + 62\right)\cdot 71 + \left(55 a + 34\right)\cdot 71^{2} + \left(48 a + 54\right)\cdot 71^{3} + \left(41 a + 35\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 68 a + 1 + \left(48 a + 33\right)\cdot 71 + \left(15 a + 52\right)\cdot 71^{2} + \left(22 a + 25\right)\cdot 71^{3} + \left(29 a + 70\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 12 a + 52 + \left(13 a + 21\right)\cdot 71 + \left(48 a + 66\right)\cdot 71^{2} + \left(65 a + 16\right)\cdot 71^{3} + \left(30 a + 32\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,5,6,3,4)$ |
| $(1,6)(2,3)(4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,6)(2,3)(4,5)$ |
$-1$ |
$-1$ |
| $1$ |
$3$ |
$(1,5,3)(2,6,4)$ |
$\zeta_{3}$ |
$-\zeta_{3} - 1$ |
| $1$ |
$3$ |
$(1,3,5)(2,4,6)$ |
$-\zeta_{3} - 1$ |
$\zeta_{3}$ |
| $1$ |
$6$ |
$(1,2,5,6,3,4)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
| $1$ |
$6$ |
$(1,4,3,6,5,2)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
