Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $ x^{3} + 6 x + 35 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 16 a^{2} + 5 a + 17 + \left(25 a^{2} + 36 a + 11\right)\cdot 37 + \left(35 a^{2} + 33 a + 30\right)\cdot 37^{2} + \left(5 a^{2} + 15 a + 20\right)\cdot 37^{3} + \left(6 a + 20\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 23 a^{2} + 30 a + 1 + \left(3 a^{2} + a\right)\cdot 37 + \left(2 a^{2} + a + 19\right)\cdot 37^{2} + \left(34 a^{2} + 10 a\right)\cdot 37^{3} + \left(27 a^{2} + 8 a + 6\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 13 a^{2} + 15 a + 5 + \left(17 a^{2} + 35 a + 16\right)\cdot 37 + \left(7 a^{2} + 3 a + 28\right)\cdot 37^{2} + \left(32 a^{2} + 24 a + 14\right)\cdot 37^{3} + \left(19 a^{2} + 22 a + 25\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 7 a^{2} + 5 a + 11 + \left(29 a^{2} + 22 a + 28\right)\cdot 37 + \left(23 a^{2} + 20 a + 31\right)\cdot 37^{2} + \left(9 a^{2} + 32 a + 13\right)\cdot 37^{3} + \left(29 a^{2} + 13 a + 11\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 8 a^{2} + 17 a + 22 + \left(31 a^{2} + 2 a + 34\right)\cdot 37 + \left(30 a^{2} + 36 a + 10\right)\cdot 37^{2} + \left(35 a^{2} + 33 a + 29\right)\cdot 37^{3} + \left(16 a^{2} + 7 a + 13\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 19 a^{2} + 14 a + 17 + \left(18 a^{2} + 6 a + 18\right)\cdot 37 + \left(32 a^{2} + 18 a + 34\right)\cdot 37^{2} + \left(5 a^{2} + 4 a + 1\right)\cdot 37^{3} + \left(34 a^{2} + 3 a + 25\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 7 a^{2} + 2 a + 11 + \left(4 a^{2} + 13 a + 2\right)\cdot 37 + \left(11 a^{2} + 15 a + 18\right)\cdot 37^{2} + \left(30 a^{2} + 31 a + 22\right)\cdot 37^{3} + \left(16 a^{2} + 14 a + 35\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 22 a^{2} + 12 a + 29 + \left(31 a^{2} + 13 a + 33\right)\cdot 37 + \left(30 a^{2} + 11 a + 27\right)\cdot 37^{2} + \left(18 a^{2} + 16 a + 16\right)\cdot 37^{3} + \left(29 a^{2} + 31 a + 6\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 33 a^{2} + 11 a + 36 + \left(23 a^{2} + 17 a + 2\right)\cdot 37 + \left(10 a^{2} + 7 a + 21\right)\cdot 37^{2} + \left(12 a^{2} + 16 a + 27\right)\cdot 37^{3} + \left(10 a^{2} + 2 a + 3\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,2,8,5,4,9,3,7,6)$ |
| $(1,3,5)(2,7,4)(6,9,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $3$ | $(1,5,3)(2,4,7)(6,8,9)$ | $\zeta_{9}^{3}$ |
| $1$ | $3$ | $(1,3,5)(2,7,4)(6,9,8)$ | $-\zeta_{9}^{3} - 1$ |
| $1$ | $9$ | $(1,2,8,5,4,9,3,7,6)$ | $\zeta_{9}$ |
| $1$ | $9$ | $(1,8,4,3,6,2,5,9,7)$ | $\zeta_{9}^{2}$ |
| $1$ | $9$ | $(1,4,6,5,7,8,3,2,9)$ | $\zeta_{9}^{4}$ |
| $1$ | $9$ | $(1,9,2,3,8,7,5,6,4)$ | $\zeta_{9}^{5}$ |
| $1$ | $9$ | $(1,7,9,5,2,6,3,4,8)$ | $-\zeta_{9}^{4} - \zeta_{9}$ |
| $1$ | $9$ | $(1,6,7,3,9,4,5,8,2)$ | $-\zeta_{9}^{5} - \zeta_{9}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.
