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^{3} + 4 x + 64 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 37 a^{2} + 37 a + 6 + \left(57 a^{2} + 48 a + 16\right)\cdot 71 + \left(11 a^{2} + 4 a + 2\right)\cdot 71^{2} + \left(27 a^{2} + 22 a + 28\right)\cdot 71^{3} + \left(61 a^{2} + 66 a + 49\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 62 a^{2} + 53 a + 51 + \left(53 a^{2} + 68 a + 59\right)\cdot 71 + \left(57 a^{2} + 17 a + 27\right)\cdot 71^{2} + \left(15 a^{2} + 23 a + 17\right)\cdot 71^{3} + \left(20 a^{2} + 12 a + 66\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 39 a^{2} + 69 a + 35 + \left(a^{2} + 33 a + 32\right)\cdot 71 + \left(58 a^{2} + 4 a + 54\right)\cdot 71^{2} + \left(40 a^{2} + 18 a + 40\right)\cdot 71^{3} + \left(24 a^{2} + 59 a + 69\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 58 a^{2} + 4 a + 64 + \left(31 a^{2} + 42 a\right)\cdot 71 + \left(a^{2} + 62 a + 67\right)\cdot 71^{2} + \left(21 a^{2} + 25 a + 54\right)\cdot 71^{3} + \left(29 a + 36\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 60 a^{2} + 70 a + 60 + \left(66 a^{2} + 20 a + 20\right)\cdot 71 + \left(53 a^{2} + 56 a + 15\right)\cdot 71^{2} + \left(16 a^{2} + 70 a + 19\right)\cdot 71^{3} + \left(4 a^{2} + 5 a + 42\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 66 a^{2} + 36 a + 36 + \left(11 a^{2} + 59 a + 36\right)\cdot 71 + \left(a^{2} + 61 a + 68\right)\cdot 71^{2} + \left(3 a^{2} + 30 a + 10\right)\cdot 71^{3} + \left(56 a^{2} + 16 a + 35\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 22 a^{2} + 14 a + 39 + \left(56 a^{2} + 31 a + 42\right)\cdot 71 + \left(11 a^{2} + 61 a + 23\right)\cdot 71^{2} + \left(34 a^{2} + 21 a + 66\right)\cdot 71^{3} + \left(50 a^{2} + 29 a + 4\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 45 a^{2} + 28 a + 20 + \left(32 a^{2} + 43 a + 24\right)\cdot 71 + \left(29 a^{2} + 23 a + 68\right)\cdot 71^{2} + \left(58 a^{2} + 3 a + 11\right)\cdot 71^{3} + \left(47 a^{2} + 29 a + 40\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 37 a^{2} + 44 a + 46 + \left(42 a^{2} + 6 a + 50\right)\cdot 71 + \left(58 a^{2} + 62 a + 27\right)\cdot 71^{2} + \left(66 a^{2} + 67 a + 34\right)\cdot 71^{3} + \left(18 a^{2} + 35 a + 10\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,9,7,3,8,4,6,5,2)$ |
| $(1,3)(2,8)(4,5)(7,9)$ |
| $(1,3,6)(2,7,4)(5,9,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $9$ | $2$ | $(1,3)(2,8)(4,5)(7,9)$ | $0$ |
| $2$ | $3$ | $(1,3,6)(2,7,4)(5,9,8)$ | $-1$ |
| $2$ | $9$ | $(1,9,7,3,8,4,6,5,2)$ | $-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
| $2$ | $9$ | $(1,7,8,6,2,9,3,4,5)$ | $\zeta_{9}^{5} + \zeta_{9}^{4}$ |
| $2$ | $9$ | $(1,8,2,3,5,7,6,9,4)$ | $-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
The blue line marks the conjugacy class containing complex conjugation.
