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