Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $ x^{2} + 33 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 31 a + 2 + \left(23 a + 21\right)\cdot 37 + \left(13 a + 33\right)\cdot 37^{2} + \left(20 a + 27\right)\cdot 37^{3} + \left(15 a + 27\right)\cdot 37^{4} + \left(8 a + 15\right)\cdot 37^{5} + \left(30 a + 36\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 15 a + 21 + \left(20 a + 33\right)\cdot 37 + \left(7 a + 32\right)\cdot 37^{2} + \left(a + 1\right)\cdot 37^{3} + \left(18 a + 7\right)\cdot 37^{4} + 13 a\cdot 37^{5} + \left(3 a + 12\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 22 a + 7 + \left(16 a + 26\right)\cdot 37 + \left(29 a + 5\right)\cdot 37^{2} + \left(35 a + 36\right)\cdot 37^{3} + \left(18 a + 3\right)\cdot 37^{4} + \left(23 a + 36\right)\cdot 37^{5} + \left(33 a + 11\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 9 + 14\cdot 37 + 35\cdot 37^{2} + 35\cdot 37^{3} + 25\cdot 37^{4} + 13\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 20 + 4\cdot 37 + 13\cdot 37^{2} + 24\cdot 37^{3} + 13\cdot 37^{4} + 24\cdot 37^{5} + 36\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 6 a + 15 + \left(13 a + 11\right)\cdot 37 + \left(23 a + 27\right)\cdot 37^{2} + \left(16 a + 21\right)\cdot 37^{3} + \left(21 a + 32\right)\cdot 37^{4} + \left(28 a + 33\right)\cdot 37^{5} + 6 a\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4,5,2,6,3)$ |
| $(2,3)(5,6)$ |
| $(3,4)(5,6)$ |
| $(1,5,6)(2,4,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $3$ | $2$ | $(1,2)(3,5)(4,6)$ | $0$ |
| $3$ | $2$ | $(1,3)(2,5)(4,6)$ | $0$ |
| $9$ | $2$ | $(3,4)(5,6)$ | $0$ |
| $2$ | $3$ | $(1,5,6)(2,4,3)$ | $-2$ |
| $2$ | $3$ | $(1,5,6)(2,3,4)$ | $-2$ |
| $4$ | $3$ | $(2,3,4)$ | $1$ |
| $6$ | $6$ | $(1,4,5,2,6,3)$ | $0$ |
| $6$ | $6$ | $(1,3,5,2,6,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
