Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $ x^{2} + 33 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 18 a + 29 + \left(30 a + 24\right)\cdot 37 + \left(36 a + 30\right)\cdot 37^{2} + \left(24 a + 18\right)\cdot 37^{3} + \left(7 a + 35\right)\cdot 37^{4} + \left(18 a + 2\right)\cdot 37^{5} + \left(3 a + 23\right)\cdot 37^{6} + \left(30 a + 17\right)\cdot 37^{7} + \left(28 a + 18\right)\cdot 37^{8} +O\left(37^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 36 a + 21 + \left(20 a + 31\right)\cdot 37 + \left(6 a + 15\right)\cdot 37^{2} + 29 a\cdot 37^{3} + \left(6 a + 1\right)\cdot 37^{4} + \left(11 a + 18\right)\cdot 37^{5} + \left(25 a + 10\right)\cdot 37^{6} + \left(12 a + 24\right)\cdot 37^{7} + \left(9 a + 24\right)\cdot 37^{8} +O\left(37^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 9 + 5\cdot 37 + 10\cdot 37^{2} + 5\cdot 37^{3} + 36\cdot 37^{4} + 20\cdot 37^{6} + 23\cdot 37^{7} + 37^{8} +O\left(37^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 29 + 31\cdot 37 + 26\cdot 37^{2} + 31\cdot 37^{3} + 36\cdot 37^{5} + 16\cdot 37^{6} + 13\cdot 37^{7} + 35\cdot 37^{8} +O\left(37^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 19 a + 9 + \left(6 a + 12\right)\cdot 37 + 6\cdot 37^{2} + \left(12 a + 18\right)\cdot 37^{3} + \left(29 a + 1\right)\cdot 37^{4} + \left(18 a + 34\right)\cdot 37^{5} + \left(33 a + 13\right)\cdot 37^{6} + \left(6 a + 19\right)\cdot 37^{7} + \left(8 a + 18\right)\cdot 37^{8} +O\left(37^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 19 a + 27 + \left(6 a + 17\right)\cdot 37 + 36\cdot 37^{2} + \left(12 a + 7\right)\cdot 37^{3} + \left(29 a + 4\right)\cdot 37^{4} + \left(18 a + 31\right)\cdot 37^{5} + \left(33 a + 18\right)\cdot 37^{6} + \left(6 a + 23\right)\cdot 37^{7} + \left(8 a + 29\right)\cdot 37^{8} +O\left(37^{ 9 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 18 a + 11 + \left(30 a + 19\right)\cdot 37 + 36 a\cdot 37^{2} + \left(24 a + 29\right)\cdot 37^{3} + \left(7 a + 32\right)\cdot 37^{4} + \left(18 a + 5\right)\cdot 37^{5} + \left(3 a + 18\right)\cdot 37^{6} + \left(30 a + 13\right)\cdot 37^{7} + \left(28 a + 7\right)\cdot 37^{8} +O\left(37^{ 9 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ a + 17 + \left(16 a + 5\right)\cdot 37 + \left(30 a + 21\right)\cdot 37^{2} + \left(7 a + 36\right)\cdot 37^{3} + \left(30 a + 35\right)\cdot 37^{4} + \left(25 a + 18\right)\cdot 37^{5} + \left(11 a + 26\right)\cdot 37^{6} + \left(24 a + 12\right)\cdot 37^{7} + \left(27 a + 12\right)\cdot 37^{8} +O\left(37^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8,5,2)(3,6,4,7)$ |
| $(1,3,5,4)(2,6,8,7)$ |
| $(1,5)(2,8)(3,4)(6,7)$ |
| $(1,5)(2,6)(7,8)$ |
| $(1,2,7)(5,8,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,8)(3,4)(6,7)$ | $-2$ |
| $12$ | $2$ | $(1,5)(2,6)(7,8)$ | $0$ |
| $8$ | $3$ | $(1,2,7)(5,8,6)$ | $-1$ |
| $6$ | $4$ | $(1,8,5,2)(3,6,4,7)$ | $0$ |
| $8$ | $6$ | $(1,8,7,5,2,6)(3,4)$ | $1$ |
| $6$ | $8$ | $(1,7,3,2,5,6,4,8)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $6$ | $8$ | $(1,6,3,8,5,7,4,2)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
