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:
| $r_{ 1 }$ |
$=$ |
$ 38 a + 44 + \left(33 a + 8\right)\cdot 71 + \left(46 a + 63\right)\cdot 71^{2} + \left(16 a + 66\right)\cdot 71^{3} + \left(32 a + 58\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 33 a + 41 + \left(37 a + 26\right)\cdot 71 + \left(24 a + 50\right)\cdot 71^{2} + \left(54 a + 5\right)\cdot 71^{3} + \left(38 a + 24\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 38 a + 36 + \left(33 a + 68\right)\cdot 71 + \left(46 a + 61\right)\cdot 71^{2} + \left(16 a + 18\right)\cdot 71^{3} + \left(32 a + 47\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 33 a + 24 + \left(37 a + 50\right)\cdot 71 + \left(24 a + 22\right)\cdot 71^{2} + \left(54 a + 27\right)\cdot 71^{3} + \left(38 a + 47\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 33 a + 49 + \left(37 a + 37\right)\cdot 71 + \left(24 a + 51\right)\cdot 71^{2} + \left(54 a + 53\right)\cdot 71^{3} + \left(38 a + 35\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 38 a + 19 + \left(33 a + 21\right)\cdot 71 + \left(46 a + 34\right)\cdot 71^{2} + \left(16 a + 40\right)\cdot 71^{3} + \left(32 a + 70\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4,3,5,6,2)$ |
| $(1,5)(2,3)(4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,5)(2,3)(4,6)$ |
$-1$ |
$-1$ |
| $1$ |
$3$ |
$(1,3,6)(2,4,5)$ |
$\zeta_{3}$ |
$-\zeta_{3} - 1$ |
| $1$ |
$3$ |
$(1,6,3)(2,5,4)$ |
$-\zeta_{3} - 1$ |
$\zeta_{3}$ |
| $1$ |
$6$ |
$(1,4,3,5,6,2)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
| $1$ |
$6$ |
$(1,2,6,5,3,4)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
