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:
| $r_{ 1 }$ |
$=$ |
$ 3 a^{2} + 62 a + 66 + \left(54 a^{2} + 30 a + 4\right)\cdot 71 + \left(33 a^{2} + 22 a + 12\right)\cdot 71^{2} + \left(5 a^{2} + 21 a + 15\right)\cdot 71^{3} + \left(45 a^{2} + 31 a + 28\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 51 a^{2} + 34 a + 52 + \left(35 a^{2} + 22 a + 50\right)\cdot 71 + \left(41 a^{2} + 70 a + 32\right)\cdot 71^{2} + \left(65 a^{2} + 25 a + 33\right)\cdot 71^{3} + \left(47 a^{2} + 54 a + 59\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 10 a^{2} + 23 a + 42 + \left(64 a^{2} + 15 a + 58\right)\cdot 71 + \left(24 a^{2} + 64 a + 38\right)\cdot 71^{2} + \left(31 a^{2} + 40 a + 68\right)\cdot 71^{3} + \left(56 a^{2} + 20 a + 1\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 29 a^{2} + 25 a + 4 + \left(40 a^{2} + 36 a + 28\right)\cdot 71 + \left(58 a^{2} + 35 a + 1\right)\cdot 71^{2} + \left(51 a^{2} + 51 a + 58\right)\cdot 71^{3} + \left(68 a^{2} + 10 a + 68\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 13 a^{2} + 9 a + 50 + \left(69 a^{2} + 62 a + 24\right)\cdot 71 + \left(37 a^{2} + 60 a + 26\right)\cdot 71^{2} + \left(33 a^{2} + 41 a + 3\right)\cdot 71^{3} + \left(48 a^{2} + 35 a + 28\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 48 a^{2} + 39 a + 25 + \left(8 a^{2} + 64 a + 5\right)\cdot 71 + \left(8 a^{2} + 16 a + 65\right)\cdot 71^{2} + \left(6 a^{2} + 59 a + 24\right)\cdot 71^{3} + \left(37 a^{2} + 14 a + 21\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 49 a + 45 + \left(18 a^{2} + 36 a + 15\right)\cdot 71 + \left(37 a^{2} + 27 a + 39\right)\cdot 71^{2} + \left(35 a^{2} + 6 a + 14\right)\cdot 71^{3} + \left(58 a^{2} + 26 a + 65\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 17 a^{2} + 46 a + 56 + \left(52 a^{2} + 17 a + 23\right)\cdot 71 + \left(66 a^{2} + 49 a + 5\right)\cdot 71^{2} + \left(70 a^{2} + 23 a + 24\right)\cdot 71^{3} + \left(48 a^{2} + 56 a + 62\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 42 a^{2} + 68 a + 15 + \left(12 a^{2} + 68 a + 1\right)\cdot 71 + \left(46 a^{2} + 7 a + 63\right)\cdot 71^{2} + \left(54 a^{2} + 13 a + 41\right)\cdot 71^{3} + \left(14 a^{2} + 34 a + 19\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,8,2)(3,6,5)(4,7,9)$ |
| $(1,6)(2,5)(3,8)(4,7)$ |
| $(1,4,3,8,7,6,2,9,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $9$ | $2$ | $(1,6)(2,5)(3,8)(4,7)$ | $0$ |
| $2$ | $3$ | $(1,8,2)(3,6,5)(4,7,9)$ | $-1$ |
| $2$ | $9$ | $(1,4,3,8,7,6,2,9,5)$ | $-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
| $2$ | $9$ | $(1,3,7,2,5,4,8,6,9)$ | $\zeta_{9}^{5} + \zeta_{9}^{4}$ |
| $2$ | $9$ | $(1,7,5,8,9,3,2,4,6)$ | $-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
The blue line marks the conjugacy class containing complex conjugation.
