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:
| $r_{ 1 }$ |
$=$ |
$ 55 a^{2} + 9 a + 27 + \left(42 a^{2} + 6 a + 31\right)\cdot 71 + \left(14 a^{2} + 65 a + 21\right)\cdot 71^{2} + \left(39 a^{2} + 64 a + 20\right)\cdot 71^{3} + \left(45 a^{2} + 69 a + 34\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 43 a^{2} + 26 a + 4 + \left(20 a^{2} + 12\right)\cdot 71 + \left(34 a^{2} + 22 a + 56\right)\cdot 71^{2} + \left(46 a^{2} + 33 a + 3\right)\cdot 71^{3} + \left(23 a^{2} + 54 a + 36\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 46 a^{2} + 49 a + 46 + \left(3 a^{2} + 53 a + 40\right)\cdot 71 + \left(13 a^{2} + 37 a + 16\right)\cdot 71^{2} + \left(62 a^{2} + 67 a + 15\right)\cdot 71^{3} + \left(38 a^{2} + 67 a + 52\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 15 a^{2} + 62 a + 15 + \left(34 a^{2} + 52 a + 32\right)\cdot 71 + \left(57 a^{2} + 24 a + 17\right)\cdot 71^{2} + \left(29 a^{2} + 3 a + 66\right)\cdot 71^{3} + \left(14 a^{2} + 7 a + 45\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 20 a^{2} + 67 a + 61 + \left(55 a^{2} + 5 a + 9\right)\cdot 71 + \left(6 a^{2} + 40 a + 30\right)\cdot 71^{2} + \left(10 a^{2} + 55 a + 1\right)\cdot 71^{3} + \left(25 a^{2} + 17 a + 40\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 3 a^{2} + 13 a + 26 + \left(25 a^{2} + 38 a + 50\right)\cdot 71 + \left(46 a^{2} + 17 a + 10\right)\cdot 71^{2} + \left(63 a^{2} + 24 a + 19\right)\cdot 71^{3} + \left(51 a^{2} + 26 a + 63\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ a^{2} + 25 + \left(65 a^{2} + 12 a + 43\right)\cdot 71 + \left(69 a^{2} + 52 a + 50\right)\cdot 71^{2} + \left(a^{2} + 2 a + 15\right)\cdot 71^{3} + \left(11 a^{2} + 65 a + 13\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 8 a^{2} + 49 a + 29 + \left(66 a^{2} + 64 a + 62\right)\cdot 71 + \left(29 a^{2} + 8 a + 20\right)\cdot 71^{2} + \left(14 a^{2} + 53 a + 60\right)\cdot 71^{3} + \left(22 a^{2} + 69 a + 55\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 22 a^{2} + 9 a + 53 + \left(42 a^{2} + 50 a + 1\right)\cdot 71 + \left(11 a^{2} + 15 a + 60\right)\cdot 71^{2} + \left(16 a^{2} + 50 a + 10\right)\cdot 71^{3} + \left(51 a^{2} + 47 a + 14\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,3,8,7,9,5,4,6,2)$ |
| $(1,6)(3,4)(5,8)(7,9)$ |
| $(1,7,4)(2,8,5)(3,9,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $9$ | $2$ | $(1,6)(3,4)(5,8)(7,9)$ | $0$ |
| $2$ | $3$ | $(1,7,4)(2,8,5)(3,9,6)$ | $-1$ |
| $2$ | $9$ | $(1,3,8,7,9,5,4,6,2)$ | $-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
| $2$ | $9$ | $(1,8,9,4,2,3,7,5,6)$ | $\zeta_{9}^{5} + \zeta_{9}^{4}$ |
| $2$ | $9$ | $(1,9,2,7,6,8,4,3,5)$ | $-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
The blue line marks the conjugacy class containing complex conjugation.
