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:
| $r_{ 1 }$ |
$=$ |
$ 58 a + 30 + \left(11 a + 64\right)\cdot 71 + \left(55 a + 40\right)\cdot 71^{2} + \left(25 a + 21\right)\cdot 71^{3} + \left(22 a + 15\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 58 a + 53 + \left(11 a + 27\right)\cdot 71 + \left(55 a + 5\right)\cdot 71^{2} + \left(25 a + 28\right)\cdot 71^{3} + \left(22 a + 26\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 13 a + 4 + \left(59 a + 30\right)\cdot 71 + \left(15 a + 68\right)\cdot 71^{2} + \left(45 a + 17\right)\cdot 71^{3} + \left(48 a + 34\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 13 a + 27 + \left(59 a + 64\right)\cdot 71 + \left(15 a + 32\right)\cdot 71^{2} + \left(45 a + 24\right)\cdot 71^{3} + \left(48 a + 45\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 13 a + 37 + \left(59 a + 31\right)\cdot 71 + \left(15 a + 46\right)\cdot 71^{2} + \left(45 a + 58\right)\cdot 71^{3} + \left(48 a + 19\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 58 a + 63 + \left(11 a + 65\right)\cdot 71 + \left(55 a + 18\right)\cdot 71^{2} + \left(25 a + 62\right)\cdot 71^{3} + 22 a\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,6)(3,4,5)$ |
| $(1,3)(2,4)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,3)(2,4)(5,6)$ | $-1$ |
| $1$ | $3$ | $(1,2,6)(3,4,5)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,6,2)(3,5,4)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,4,6,3,2,5)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,5,2,3,6,4)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
