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:
| $r_{ 1 }$ |
$=$ |
$ 64 a + 30 + \left(13 a + 34\right)\cdot 71 + \left(48 a + 57\right)\cdot 71^{2} + \left(15 a + 53\right)\cdot 71^{3} + \left(33 a + 56\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 a + 16 + \left(57 a + 69\right)\cdot 71 + \left(22 a + 68\right)\cdot 71^{2} + \left(55 a + 36\right)\cdot 71^{3} + \left(37 a + 36\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 68 a + 66 + \left(12 a + 16\right)\cdot 71 + \left(2 a + 14\right)\cdot 71^{2} + \left(14 a + 37\right)\cdot 71^{3} + \left(18 a + 70\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 69 + 64\cdot 71 + 11\cdot 71^{2} + 58\cdot 71^{3} + 58\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 43 + 52\cdot 71 + 54\cdot 71^{2} + 34\cdot 71^{3} + 39\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 3 a + 60 + \left(58 a + 45\right)\cdot 71 + \left(68 a + 5\right)\cdot 71^{2} + \left(56 a + 63\right)\cdot 71^{3} + \left(52 a + 21\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)$ |
| $(4,5)$ |
| $(3,6)$ |
| $(1,4,3)(2,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,2)(3,6)(4,5)$ | $-3$ |
| $3$ | $2$ | $(3,6)$ | $1$ |
| $3$ | $2$ | $(1,2)(3,6)$ | $-1$ |
| $4$ | $3$ | $(1,4,3)(2,5,6)$ | $0$ |
| $4$ | $3$ | $(1,3,4)(2,6,5)$ | $0$ |
| $4$ | $6$ | $(1,4,3,2,5,6)$ | $0$ |
| $4$ | $6$ | $(1,6,5,2,3,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
