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:
| $r_{ 1 }$ |
$=$ |
$ 31 a^{2} + 26 a + 18 + \left(19 a + 16\right)\cdot 37 + \left(15 a^{2} + 27 a + 5\right)\cdot 37^{2} + \left(32 a^{2} + 13 a + 31\right)\cdot 37^{3} + \left(8 a^{2} + 30 a + 9\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 25 a^{2} + 33 a + 6 + \left(35 a^{2} + 26 a + 16\right)\cdot 37 + \left(20 a + 11\right)\cdot 37^{2} + \left(23 a^{2} + 29 a + 32\right)\cdot 37^{3} + \left(14 a^{2} + 36 a + 6\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 20 a^{2} + 5 a + 23 + \left(24 a^{2} + 32 a + 8\right)\cdot 37 + \left(35 a^{2} + a + 2\right)\cdot 37^{2} + \left(10 a^{2} + 21\right)\cdot 37^{3} + \left(6 a^{2} + 19 a + 10\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 26 a^{2} + 9 a + 34 + \left(4 a^{2} + 8 a + 8\right)\cdot 37 + \left(15 a^{2} + 18 a + 21\right)\cdot 37^{2} + \left(23 a^{2} + 32 a + 17\right)\cdot 37^{3} + \left(5 a^{2} + 26 a + 13\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 22 a^{2} + 17 a + 19 + \left(4 a^{2} + 10 a + 31\right)\cdot 37 + \left(2 a^{2} + 29 a + 27\right)\cdot 37^{2} + \left(28 a^{2} + 13\right)\cdot 37^{3} + \left(28 a^{2} + 21 a + 15\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 15 a^{2} + 7 a + 27 + \left(3 a^{2} + 3\right)\cdot 37 + \left(36 a^{2} + 8 a + 31\right)\cdot 37^{2} + \left(25 a^{2} + 33 a + 27\right)\cdot 37^{3} + \left(28 a^{2} + 29 a + 31\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 29 a^{2} + 36 a + 22 + \left(13 a^{2} + 14 a + 2\right)\cdot 37 + \left(14 a + 9\right)\cdot 37^{2} + \left(3 a^{2} + 7 a + 26\right)\cdot 37^{3} + \left(16 a^{2} + 18 a + 12\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 21 a^{2} + 31 a + 15 + \left(31 a^{2} + 6 a + 28\right)\cdot 37 + \left(19 a^{2} + 17 a + 24\right)\cdot 37^{2} + \left(13 a^{2} + 22 a + 29\right)\cdot 37^{3} + \left(36 a^{2} + 22 a + 8\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 33 a^{2} + 21 a + 25 + \left(28 a^{2} + 28 a + 31\right)\cdot 37 + \left(22 a^{2} + 10 a + 14\right)\cdot 37^{2} + \left(24 a^{2} + 8 a + 22\right)\cdot 37^{3} + \left(2 a^{2} + 17 a + 1\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,8,5)(2,3,7)(4,9,6)$ |
| $(1,5)(2,4)(3,6)(7,9)$ |
| $(1,2,6,8,3,4,5,7,9)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $9$ | $2$ | $(1,5)(2,4)(3,6)(7,9)$ | $0$ |
| $2$ | $3$ | $(1,8,5)(2,3,7)(4,9,6)$ | $-1$ |
| $2$ | $9$ | $(1,2,6,8,3,4,5,7,9)$ | $\zeta_{9}^{5} + \zeta_{9}^{4}$ |
| $2$ | $9$ | $(1,6,3,5,9,2,8,4,7)$ | $-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
| $2$ | $9$ | $(1,3,9,8,7,6,5,2,4)$ | $-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
The blue line marks the conjugacy class containing complex conjugation.
