Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 71 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 71 }$: $ x^{2} + 69 x + 7 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 a + 67 + \left(61 a + 11\right)\cdot 71 + \left(a + 64\right)\cdot 71^{2} + \left(51 a + 20\right)\cdot 71^{3} + \left(52 a + 8\right)\cdot 71^{4} + \left(46 a + 15\right)\cdot 71^{5} + \left(18 a + 40\right)\cdot 71^{6} +O\left(71^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 a + 68 + \left(66 a + 41\right)\cdot 71 + \left(56 a + 11\right)\cdot 71^{2} + \left(28 a + 35\right)\cdot 71^{3} + \left(19 a + 30\right)\cdot 71^{4} + \left(36 a + 44\right)\cdot 71^{5} + \left(46 a + 42\right)\cdot 71^{6} +O\left(71^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 64 a + 7 + \left(14 a + 17\right)\cdot 71 + \left(12 a + 66\right)\cdot 71^{2} + \left(62 a + 14\right)\cdot 71^{3} + \left(69 a + 32\right)\cdot 71^{4} + \left(58 a + 11\right)\cdot 71^{5} + \left(5 a + 59\right)\cdot 71^{6} +O\left(71^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 67 a + 4 + \left(9 a + 59\right)\cdot 71 + \left(69 a + 6\right)\cdot 71^{2} + \left(19 a + 50\right)\cdot 71^{3} + \left(18 a + 62\right)\cdot 71^{4} + \left(24 a + 55\right)\cdot 71^{5} + \left(52 a + 30\right)\cdot 71^{6} +O\left(71^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 68 a + 3 + \left(4 a + 29\right)\cdot 71 + \left(14 a + 59\right)\cdot 71^{2} + \left(42 a + 35\right)\cdot 71^{3} + \left(51 a + 40\right)\cdot 71^{4} + \left(34 a + 26\right)\cdot 71^{5} + \left(24 a + 28\right)\cdot 71^{6} +O\left(71^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 7 a + 64 + \left(56 a + 53\right)\cdot 71 + \left(58 a + 4\right)\cdot 71^{2} + \left(8 a + 56\right)\cdot 71^{3} + \left(a + 38\right)\cdot 71^{4} + \left(12 a + 59\right)\cdot 71^{5} + \left(65 a + 11\right)\cdot 71^{6} +O\left(71^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)(2,5)(3,6)$ |
| $(1,2,3)(4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,4)(2,5)(3,6)$ |
$-1$ |
$-1$ |
| $1$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$\zeta_{3}$ |
$-\zeta_{3} - 1$ |
| $1$ |
$3$ |
$(1,3,2)(4,6,5)$ |
$-\zeta_{3} - 1$ |
$\zeta_{3}$ |
| $1$ |
$6$ |
$(1,5,3,4,2,6)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
| $1$ |
$6$ |
$(1,6,2,4,3,5)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
