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(8 a + 3\right)\cdot 37 + \left(27 a + 31\right)\cdot 37^{2} + \left(9 a + 4\right)\cdot 37^{3} + \left(2 a + 21\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 13 + 27\cdot 37 + 11\cdot 37^{2} + 24\cdot 37^{3} + 16\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 a + 3 + \left(6 a + 27\right)\cdot 37 + \left(a + 7\right)\cdot 37^{2} + \left(22 a + 7\right)\cdot 37^{3} + \left(18 a + 24\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 19 a + 1 + \left(30 a + 35\right)\cdot 37 + \left(35 a + 5\right)\cdot 37^{2} + \left(14 a + 20\right)\cdot 37^{3} + \left(18 a + 2\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 26 a + 28 + 4\cdot 37 + 27 a\cdot 37^{2} + \left(12 a + 7\right)\cdot 37^{3} + \left(14 a + 9\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 11 a + 21 + \left(36 a + 18\right)\cdot 37 + \left(9 a + 33\right)\cdot 37^{2} + \left(24 a + 30\right)\cdot 37^{3} + \left(22 a + 16\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 32 a + 15 + \left(28 a + 31\right)\cdot 37 + \left(9 a + 20\right)\cdot 37^{2} + \left(27 a + 16\right)\cdot 37^{3} + \left(34 a + 20\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 7 }$
| Cycle notation |
| $(1,6)(2,7)(4,5)$ |
| $(1,3)(2,4)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 7 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $7$ | $2$ | $(1,3)(2,4)(6,7)$ | $0$ |
| $2$ | $7$ | $(1,3,6,2,5,4,7)$ | $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - 1$ |
| $2$ | $7$ | $(1,6,5,7,3,2,4)$ | $\zeta_{7}^{5} + \zeta_{7}^{2}$ |
| $2$ | $7$ | $(1,2,7,6,4,3,5)$ | $\zeta_{7}^{4} + \zeta_{7}^{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
