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 + 13 + \left(52 a + 14\right)\cdot 71 + \left(63 a + 53\right)\cdot 71^{2} + \left(27 a + 68\right)\cdot 71^{3} + \left(10 a + 24\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 64 a + 18 + \left(18 a + 10\right)\cdot 71 + \left(7 a + 22\right)\cdot 71^{2} + \left(43 a + 51\right)\cdot 71^{3} + \left(60 a + 18\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 64 a + 27 + \left(18 a + 40\right)\cdot 71 + \left(7 a + 57\right)\cdot 71^{2} + \left(43 a + 60\right)\cdot 71^{3} + \left(60 a + 17\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 7 a + 4 + \left(52 a + 55\right)\cdot 71 + \left(63 a + 17\right)\cdot 71^{2} + \left(27 a + 59\right)\cdot 71^{3} + \left(10 a + 25\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 7 a + 70 + \left(52 a + 68\right)\cdot 71 + \left(63 a + 28\right)\cdot 71^{2} + \left(27 a + 61\right)\cdot 71^{3} + \left(10 a + 30\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 64 a + 13 + \left(18 a + 24\right)\cdot 71 + \left(7 a + 33\right)\cdot 71^{2} + \left(43 a + 53\right)\cdot 71^{3} + \left(60 a + 23\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(2,4)(5,6)$ |
| $(1,2,5,3,4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,3)(2,4)(5,6)$ |
$-1$ |
$-1$ |
| $1$ |
$3$ |
$(1,5,4)(2,3,6)$ |
$\zeta_{3}$ |
$-\zeta_{3} - 1$ |
| $1$ |
$3$ |
$(1,4,5)(2,6,3)$ |
$-\zeta_{3} - 1$ |
$\zeta_{3}$ |
| $1$ |
$6$ |
$(1,2,5,3,4,6)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
| $1$ |
$6$ |
$(1,6,4,3,5,2)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
