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:
| $r_{ 1 }$ |
$=$ |
$ 38 + 52\cdot 71 + 12\cdot 71^{2} + 20\cdot 71^{3} + 23\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 58 a + 53 + \left(29 a + 18\right)\cdot 71 + 20 a\cdot 71^{2} + \left(15 a + 34\right)\cdot 71^{3} + \left(34 a + 27\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 22 + 42\cdot 71 + 40\cdot 71^{2} + 17\cdot 71^{3} + 42\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 13 a + 27 + \left(41 a + 20\right)\cdot 71 + \left(50 a + 11\right)\cdot 71^{2} + \left(55 a + 44\right)\cdot 71^{3} + \left(36 a + 9\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 36 a + 1 + \left(44 a + 13\right)\cdot 71 + \left(29 a + 31\right)\cdot 71^{2} + \left(67 a + 31\right)\cdot 71^{3} + \left(66 a + 57\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 35 a + 2 + \left(26 a + 66\right)\cdot 71 + \left(41 a + 45\right)\cdot 71^{2} + \left(3 a + 65\right)\cdot 71^{3} + \left(4 a + 52\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $9$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $3$ |
| $15$ | $2$ | $(1,2)$ | $3$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $40$ | $3$ | $(1,2,3)$ | $0$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $1$ |
| $90$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
