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:
| $r_{ 1 }$ |
$=$ |
$ 36 a + 9 + \left(67 a + 51\right)\cdot 71 + \left(8 a + 50\right)\cdot 71^{2} + \left(14 a + 33\right)\cdot 71^{3} + \left(12 a + 53\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 35 a + 10 + \left(3 a + 8\right)\cdot 71 + \left(62 a + 1\right)\cdot 71^{2} + \left(56 a + 53\right)\cdot 71^{3} + \left(58 a + 63\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 33 a + 31 + \left(65 a + 15\right)\cdot 71 + \left(17 a + 16\right)\cdot 71^{2} + \left(66 a + 1\right)\cdot 71^{3} + \left(30 a + 68\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 38 a + 26 + \left(5 a + 42\right)\cdot 71 + \left(53 a + 57\right)\cdot 71^{2} + \left(4 a + 44\right)\cdot 71^{3} + \left(40 a + 63\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 45 a + 24 + \left(69 a + 36\right)\cdot 71 + \left(24 a + 53\right)\cdot 71^{2} + \left(22 a + 65\right)\cdot 71^{3} + \left(a + 62\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 26 a + 43 + \left(a + 59\right)\cdot 71 + \left(46 a + 33\right)\cdot 71^{2} + \left(48 a + 14\right)\cdot 71^{3} + \left(69 a + 43\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)$ |
| $(1,6,3)(2,5,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$-1$ |
$-1$ |
| $1$ |
$3$ |
$(1,6,3)(2,5,4)$ |
$\zeta_{3}$ |
$-\zeta_{3} - 1$ |
| $1$ |
$3$ |
$(1,3,6)(2,4,5)$ |
$-\zeta_{3} - 1$ |
$\zeta_{3}$ |
| $1$ |
$6$ |
$(1,5,3,2,6,4)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
| $1$ |
$6$ |
$(1,4,6,2,3,5)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
