Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$: $ x^{2} + 70 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 72 a + 20 + \left(71 a + 1\right)\cdot 73 + \left(15 a + 30\right)\cdot 73^{2} + \left(69 a + 68\right)\cdot 73^{3} + \left(67 a + 43\right)\cdot 73^{4} + \left(49 a + 19\right)\cdot 73^{5} + \left(12 a + 7\right)\cdot 73^{6} + \left(65 a + 45\right)\cdot 73^{7} +O\left(73^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 72 a + 23 + \left(71 a + 26\right)\cdot 73 + \left(15 a + 35\right)\cdot 73^{2} + \left(69 a + 37\right)\cdot 73^{3} + \left(67 a + 5\right)\cdot 73^{4} + \left(49 a + 2\right)\cdot 73^{5} + \left(12 a + 45\right)\cdot 73^{6} + \left(65 a + 9\right)\cdot 73^{7} +O\left(73^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ a + 17 + \left(a + 72\right)\cdot 73 + \left(57 a + 5\right)\cdot 73^{2} + \left(3 a + 41\right)\cdot 73^{3} + \left(5 a + 32\right)\cdot 73^{4} + \left(23 a + 28\right)\cdot 73^{5} + \left(60 a + 68\right)\cdot 73^{6} + \left(7 a + 8\right)\cdot 73^{7} +O\left(73^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 35 + 23\cdot 73 + 34\cdot 73^{2} + 15\cdot 73^{3} + 52\cdot 73^{4} + 33\cdot 73^{5} + 51\cdot 73^{6} + 36\cdot 73^{7} +O\left(73^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 32 + 71\cdot 73 + 28\cdot 73^{2} + 46\cdot 73^{3} + 17\cdot 73^{4} + 51\cdot 73^{5} + 13\cdot 73^{6} + 72\cdot 73^{7} +O\left(73^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ a + 20 + \left(a + 24\right)\cdot 73 + \left(57 a + 11\right)\cdot 73^{2} + \left(3 a + 10\right)\cdot 73^{3} + \left(5 a + 67\right)\cdot 73^{4} + \left(23 a + 10\right)\cdot 73^{5} + \left(60 a + 33\right)\cdot 73^{6} + \left(7 a + 46\right)\cdot 73^{7} +O\left(73^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3,5)(2,6,4)$ |
| $(3,5)(4,6)$ |
| $(1,2)(3,6)(4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,6)(4,5)$ | $-2$ |
| $3$ | $2$ | $(3,5)(4,6)$ | $0$ |
| $3$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $2$ | $3$ | $(1,3,5)(2,6,4)$ | $-1$ |
| $2$ | $6$ | $(1,6,5,2,3,4)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
