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:
| $r_{ 1 }$ |
$=$ |
$ 63 a + 53 + \left(39 a + 56\right)\cdot 71 + \left(12 a + 68\right)\cdot 71^{2} + \left(35 a + 49\right)\cdot 71^{3} + \left(31 a + 44\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 61 a + 8 + \left(a + 41\right)\cdot 71 + \left(54 a + 61\right)\cdot 71^{2} + \left(57 a + 44\right)\cdot 71^{3} + \left(65 a + 51\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 10 a + 59 + \left(69 a + 54\right)\cdot 71 + \left(16 a + 25\right)\cdot 71^{2} + \left(13 a + 35\right)\cdot 71^{3} + \left(5 a + 54\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 40 a + 24 + \left(26 a + 22\right)\cdot 71 + \left(25 a + 60\right)\cdot 71^{2} + \left(65 a + 5\right)\cdot 71^{3} + \left(7 a + 55\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 8 a + 37 + \left(31 a + 2\right)\cdot 71 + \left(58 a + 54\right)\cdot 71^{2} + \left(35 a + 36\right)\cdot 71^{3} + \left(39 a + 1\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 31 a + 33 + \left(44 a + 35\right)\cdot 71 + \left(45 a + 13\right)\cdot 71^{2} + \left(5 a + 40\right)\cdot 71^{3} + \left(63 a + 5\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,6,3,5,4,2)$ |
| $(1,5)(2,3)(4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,5)(2,3)(4,6)$ |
$-1$ |
$-1$ |
| $1$ |
$3$ |
$(1,3,4)(2,6,5)$ |
$\zeta_{3}$ |
$-\zeta_{3} - 1$ |
| $1$ |
$3$ |
$(1,4,3)(2,5,6)$ |
$-\zeta_{3} - 1$ |
$\zeta_{3}$ |
| $1$ |
$6$ |
$(1,6,3,5,4,2)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
| $1$ |
$6$ |
$(1,2,4,5,3,6)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
