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:
| $r_{ 1 }$ |
$=$ |
$ 27 + 38\cdot 71 + 53\cdot 71^{2} + 56\cdot 71^{3} + 55\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 55 a + 69 + \left(69 a + 50\right)\cdot 71 + \left(34 a + 62\right)\cdot 71^{2} + \left(22 a + 30\right)\cdot 71^{3} + \left(2 a + 69\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 51 a + 55 + \left(34 a + 38\right)\cdot 71 + \left(28 a + 5\right)\cdot 71^{2} + \left(22 a + 45\right)\cdot 71^{3} + \left(11 a + 29\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 21 a + 55 + 62\cdot 71 + \left(12 a + 23\right)\cdot 71^{2} + \left(44 a + 57\right)\cdot 71^{3} + \left(26 a + 54\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 20 a + 15 + \left(36 a + 57\right)\cdot 71 + \left(42 a + 27\right)\cdot 71^{2} + \left(48 a + 61\right)\cdot 71^{3} + \left(59 a + 29\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 16 a + 37 + \left(a + 64\right)\cdot 71 + \left(36 a + 62\right)\cdot 71^{2} + \left(48 a + 40\right)\cdot 71^{3} + \left(68 a + 51\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 50 a + 26 + \left(70 a + 42\right)\cdot 71 + \left(58 a + 47\right)\cdot 71^{2} + \left(26 a + 62\right)\cdot 71^{3} + \left(44 a + 63\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 7 }$
| Cycle notation |
| $(1,2,3,4,5,6,7)$ |
| $(1,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 7 }$
| Character value |
| $1$ | $1$ | $()$ | $6$ |
| $21$ | $2$ | $(1,2)$ | $-4$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $2$ |
| $70$ | $3$ | $(1,2,3)$ | $3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $210$ | $4$ | $(1,2,3,4)$ | $-2$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $-1$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
