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^{3} + 6 x + 35 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 24 a^{2} + 16 a + 3 + \left(27 a^{2} + 8 a + 25\right)\cdot 37 + \left(27 a^{2} + 23 a + 28\right)\cdot 37^{2} + \left(6 a^{2} + 36 a + 5\right)\cdot 37^{3} + \left(13 a + 18\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 15 + 7\cdot 37 + 6\cdot 37^{2} + 26\cdot 37^{3} +O\left(37^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 33 a^{2} + 29 a + 36 + \left(29 a^{2} + 2 a + 29\right)\cdot 37 + \left(15 a^{2} + 3 a + 7\right)\cdot 37^{2} + \left(34 a^{2} + 3 a + 2\right)\cdot 37^{3} + \left(12 a^{2} + 7 a + 34\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 34 a^{2} + 7 a + 6 + \left(18 a^{2} + 14 a + 27\right)\cdot 37 + \left(32 a^{2} + 19 a + 10\right)\cdot 37^{2} + \left(14 a^{2} + 14 a + 1\right)\cdot 37^{3} + \left(a^{2} + 31 a + 23\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 7 a^{2} + 14 a + 6 + \left(3 a^{2} + 34\right)\cdot 37 + \left(6 a^{2} + 12 a + 5\right)\cdot 37^{2} + \left(2 a + 13\right)\cdot 37^{3} + \left(3 a^{2} + 28 a + 31\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 16 a^{2} + 14 a + 8 + \left(27 a^{2} + 14 a + 24\right)\cdot 37 + \left(13 a^{2} + 31 a + 9\right)\cdot 37^{2} + \left(15 a^{2} + 22 a + 3\right)\cdot 37^{3} + \left(35 a^{2} + 28 a + 11\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 34 a^{2} + 31 a + 3 + \left(3 a^{2} + 33 a\right)\cdot 37 + \left(15 a^{2} + 21 a + 5\right)\cdot 37^{2} + \left(2 a^{2} + 31 a + 22\right)\cdot 37^{3} + \left(21 a^{2} + a + 29\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 7 }$
| Cycle notation |
| $(2,4)(5,7)$ |
| $(1,4)(2,7,6,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 7 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$3$ |
$3$ |
| $21$ |
$2$ |
$(2,6)(3,7)$ |
$-1$ |
$-1$ |
| $56$ |
$3$ |
$(1,7,3)(2,5,6)$ |
$0$ |
$0$ |
| $42$ |
$4$ |
$(1,4)(2,7,6,3)$ |
$1$ |
$1$ |
| $24$ |
$7$ |
$(1,2,5,7,6,3,4)$ |
$\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$ |
$-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ |
| $24$ |
$7$ |
$(1,7,4,5,3,2,6)$ |
$-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ |
$\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$ |
The blue line marks the conjugacy class containing complex conjugation.
