Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $ x^{2} + 33 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 10 a + 29 + \left(17 a + 30\right)\cdot 37 + \left(33 a + 21\right)\cdot 37^{2} + \left(26 a + 34\right)\cdot 37^{3} + \left(31 a + 17\right)\cdot 37^{4} + \left(25 a + 7\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 30 + 34\cdot 37 + 28\cdot 37^{2} + 5\cdot 37^{3} + 11\cdot 37^{4} + 14\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 27 a + 32 + \left(19 a + 15\right)\cdot 37 + \left(3 a + 27\right)\cdot 37^{2} + \left(10 a + 34\right)\cdot 37^{3} + \left(5 a + 6\right)\cdot 37^{4} + \left(11 a + 5\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 12 a + 22 + 18\cdot 37 + \left(21 a + 4\right)\cdot 37^{2} + \left(10 a + 7\right)\cdot 37^{3} + 34 a\cdot 37^{4} + \left(21 a + 29\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 20 a + 30 + \left(21 a + 24\right)\cdot 37 + \left(27 a + 35\right)\cdot 37^{2} + \left(22 a + 36\right)\cdot 37^{3} + \left(6 a + 8\right)\cdot 37^{4} + \left(14 a + 24\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 17 a + 36 + \left(15 a + 16\right)\cdot 37 + \left(9 a + 13\right)\cdot 37^{2} + \left(14 a + 26\right)\cdot 37^{3} + \left(30 a + 12\right)\cdot 37^{4} + 22 a\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 25 a + 33 + \left(36 a + 7\right)\cdot 37 + \left(15 a + 14\right)\cdot 37^{2} + \left(26 a + 28\right)\cdot 37^{3} + \left(2 a + 15\right)\cdot 37^{4} + \left(15 a + 8\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 14 + 35\cdot 37 + 37^{2} + 11\cdot 37^{3} + 22\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(2,6,4)(3,8,5)$ |
| $(1,7)(2,8)(3,4)(5,6)$ |
| $(2,8)(3,6)(4,5)$ |
| $(1,2,7,8)(3,5,4,6)$ |
| $(1,3,7,4)(2,6,8,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,8)(3,4)(5,6)$ | $-2$ |
| $12$ | $2$ | $(2,8)(3,6)(4,5)$ | $0$ |
| $8$ | $3$ | $(2,6,4)(3,8,5)$ | $-1$ |
| $6$ | $4$ | $(1,3,7,4)(2,6,8,5)$ | $0$ |
| $8$ | $6$ | $(1,3,5,7,4,6)(2,8)$ | $1$ |
| $6$ | $8$ | $(1,6,2,3,7,5,8,4)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
| $6$ | $8$ | $(1,5,2,4,7,6,8,3)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
