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:
| $r_{ 1 }$ |
$=$ |
$ 58 a + 6 + \left(51 a + 63\right)\cdot 71 + \left(38 a + 9\right)\cdot 71^{2} + \left(27 a + 22\right)\cdot 71^{3} + \left(50 a + 21\right)\cdot 71^{4} + \left(4 a + 60\right)\cdot 71^{5} + \left(2 a + 34\right)\cdot 71^{6} +O\left(71^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 42 a + 61 + \left(8 a + 25\right)\cdot 71 + \left(63 a + 33\right)\cdot 71^{2} + 20 a\cdot 71^{3} + 5 a\cdot 71^{4} + \left(22 a + 29\right)\cdot 71^{5} + \left(64 a + 2\right)\cdot 71^{6} +O\left(71^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 13 a + 51 + \left(19 a + 37\right)\cdot 71 + \left(32 a + 35\right)\cdot 71^{2} + \left(43 a + 38\right)\cdot 71^{3} + \left(20 a + 23\right)\cdot 71^{4} + \left(66 a + 19\right)\cdot 71^{5} + \left(68 a + 34\right)\cdot 71^{6} +O\left(71^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 44 + 6\cdot 71 + 57\cdot 71^{2} + 65\cdot 71^{3} + 50\cdot 71^{4} + 52\cdot 71^{5} + 33\cdot 71^{6} +O\left(71^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 50 + 7\cdot 71 + 68\cdot 71^{2} + 35\cdot 71^{3} + 56\cdot 71^{4} + 54\cdot 71^{5} + 69\cdot 71^{6} +O\left(71^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 29 a + 3 + \left(62 a + 1\right)\cdot 71 + \left(7 a + 9\right)\cdot 71^{2} + \left(50 a + 50\right)\cdot 71^{3} + \left(65 a + 60\right)\cdot 71^{4} + \left(48 a + 67\right)\cdot 71^{5} + \left(6 a + 37\right)\cdot 71^{6} +O\left(71^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3,4)(2,6,5)$ |
| $(1,4)(2,5)$ |
| $(4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,2)(3,6)(4,5)$ |
$-3$ |
| $3$ |
$2$ |
$(1,2)(4,5)$ |
$-1$ |
| $3$ |
$2$ |
$(1,2)$ |
$1$ |
| $6$ |
$2$ |
$(1,4)(2,5)$ |
$-1$ |
| $6$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$1$ |
| $8$ |
$3$ |
$(1,3,4)(2,6,5)$ |
$0$ |
| $6$ |
$4$ |
$(1,5,2,4)$ |
$-1$ |
| $6$ |
$4$ |
$(1,5,2,4)(3,6)$ |
$1$ |
| $8$ |
$6$ |
$(1,5,6,2,4,3)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
