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