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:
| $r_{ 1 }$ |
$=$ |
$ 7 a + 33 + \left(30 a + 19\right)\cdot 71 + \left(42 a + 27\right)\cdot 71^{2} + \left(53 a + 29\right)\cdot 71^{3} + \left(29 a + 68\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 64 a + 57 + \left(40 a + 39\right)\cdot 71 + \left(28 a + 24\right)\cdot 71^{2} + \left(17 a + 57\right)\cdot 71^{3} + \left(41 a + 48\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 7 a + 43 + \left(30 a + 57\right)\cdot 71 + \left(42 a + 40\right)\cdot 71^{2} + \left(53 a + 63\right)\cdot 71^{3} + \left(29 a + 42\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 64 a + 47 + \left(40 a + 1\right)\cdot 71 + \left(28 a + 11\right)\cdot 71^{2} + \left(17 a + 23\right)\cdot 71^{3} + \left(41 a + 3\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 64 a + 24 + \left(40 a + 38\right)\cdot 71 + \left(28 a + 46\right)\cdot 71^{2} + \left(17 a + 16\right)\cdot 71^{3} + \left(41 a + 63\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 7 a + 10 + \left(30 a + 56\right)\cdot 71 + \left(42 a + 62\right)\cdot 71^{2} + \left(53 a + 22\right)\cdot 71^{3} + \left(29 a + 57\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)(2,3)(5,6)$ |
| $(1,2,6,4,3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,4)(2,3)(5,6)$ |
$-1$ |
$-1$ |
| $1$ |
$3$ |
$(1,6,3)(2,4,5)$ |
$\zeta_{3}$ |
$-\zeta_{3} - 1$ |
| $1$ |
$3$ |
$(1,3,6)(2,5,4)$ |
$-\zeta_{3} - 1$ |
$\zeta_{3}$ |
| $1$ |
$6$ |
$(1,2,6,4,3,5)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
| $1$ |
$6$ |
$(1,5,3,4,6,2)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
