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