Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$: $ x^{2} + 70 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 64 a + 32 + \left(71 a + 32\right)\cdot 73 + \left(61 a + 23\right)\cdot 73^{2} + \left(28 a + 44\right)\cdot 73^{3} + \left(42 a + 15\right)\cdot 73^{4} + \left(28 a + 4\right)\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 60 a + 16 + \left(19 a + 2\right)\cdot 73 + \left(71 a + 30\right)\cdot 73^{2} + \left(15 a + 1\right)\cdot 73^{3} + \left(16 a + 68\right)\cdot 73^{4} + \left(18 a + 68\right)\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 9 a + 5 + \left(a + 38\right)\cdot 73 + \left(11 a + 64\right)\cdot 73^{2} + \left(44 a + 68\right)\cdot 73^{3} + \left(30 a + 40\right)\cdot 73^{4} + \left(44 a + 47\right)\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 4 a + 52 + \left(52 a + 32\right)\cdot 73 + \left(63 a + 51\right)\cdot 73^{2} + \left(12 a + 2\right)\cdot 73^{3} + \left(26 a + 37\right)\cdot 73^{4} + \left(10 a + 29\right)\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 69 a + 64 + \left(20 a + 38\right)\cdot 73 + \left(9 a + 44\right)\cdot 73^{2} + \left(60 a + 50\right)\cdot 73^{3} + \left(46 a + 29\right)\cdot 73^{4} + \left(62 a + 34\right)\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 13 a + 50 + \left(53 a + 1\right)\cdot 73 + \left(a + 5\right)\cdot 73^{2} + \left(57 a + 51\right)\cdot 73^{3} + \left(56 a + 27\right)\cdot 73^{4} + \left(54 a + 34\right)\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4,6,3,5,2)$ |
| $(2,3,4)$ |
| $(1,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,3)(2,6)(4,5)$ | $0$ |
| $1$ | $3$ | $(1,6,5)(2,4,3)$ | $-2 \zeta_{3} - 2$ |
| $1$ | $3$ | $(1,5,6)(2,3,4)$ | $2 \zeta_{3}$ |
| $2$ | $3$ | $(2,3,4)$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(2,4,3)$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(1,5,6)(2,4,3)$ | $-1$ |
| $3$ | $6$ | $(1,4,6,3,5,2)$ | $0$ |
| $3$ | $6$ | $(1,2,5,3,6,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
