Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 11.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$: $ x^{2} + 70 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 38 a + 31 + \left(58 a + 58\right)\cdot 73 + \left(64 a + 57\right)\cdot 73^{2} + \left(19 a + 70\right)\cdot 73^{3} + \left(30 a + 62\right)\cdot 73^{4} + \left(34 a + 13\right)\cdot 73^{5} + \left(11 a + 47\right)\cdot 73^{6} + \left(32 a + 68\right)\cdot 73^{7} + \left(59 a + 68\right)\cdot 73^{8} + \left(a + 3\right)\cdot 73^{9} + \left(60 a + 32\right)\cdot 73^{10} +O\left(73^{ 11 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 52 a + 19 + \left(39 a + 37\right)\cdot 73 + \left(71 a + 52\right)\cdot 73^{2} + \left(6 a + 13\right)\cdot 73^{3} + \left(54 a + 71\right)\cdot 73^{4} + \left(67 a + 6\right)\cdot 73^{5} + \left(3 a + 34\right)\cdot 73^{6} + \left(29 a + 25\right)\cdot 73^{7} + \left(50 a + 61\right)\cdot 73^{8} + \left(33 a + 21\right)\cdot 73^{9} + \left(63 a + 38\right)\cdot 73^{10} +O\left(73^{ 11 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 17 + 68\cdot 73 + 13\cdot 73^{2} + 30\cdot 73^{3} + 35\cdot 73^{4} + 54\cdot 73^{5} + 37\cdot 73^{6} + 47\cdot 73^{7} + 27\cdot 73^{8} + 39\cdot 73^{9} + 48\cdot 73^{10} +O\left(73^{ 11 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 35 a + 72 + \left(14 a + 49\right)\cdot 73 + \left(8 a + 47\right)\cdot 73^{2} + \left(53 a + 65\right)\cdot 73^{3} + \left(42 a + 60\right)\cdot 73^{4} + \left(38 a + 13\right)\cdot 73^{5} + \left(61 a + 47\right)\cdot 73^{6} + \left(40 a + 7\right)\cdot 73^{7} + \left(13 a + 69\right)\cdot 73^{8} + \left(71 a + 22\right)\cdot 73^{9} + \left(12 a + 64\right)\cdot 73^{10} +O\left(73^{ 11 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 21 a + 29 + \left(33 a + 31\right)\cdot 73 + \left(a + 8\right)\cdot 73^{2} + \left(66 a + 36\right)\cdot 73^{3} + \left(18 a + 7\right)\cdot 73^{4} + \left(5 a + 10\right)\cdot 73^{5} + \left(69 a + 51\right)\cdot 73^{6} + \left(43 a + 35\right)\cdot 73^{7} + \left(22 a + 37\right)\cdot 73^{8} + \left(39 a + 72\right)\cdot 73^{9} + \left(9 a + 48\right)\cdot 73^{10} +O\left(73^{ 11 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 53 + 46\cdot 73 + 38\cdot 73^{2} + 2\cdot 73^{3} + 54\cdot 73^{4} + 46\cdot 73^{5} + 73^{6} + 34\cdot 73^{7} + 27\cdot 73^{8} + 58\cdot 73^{9} + 59\cdot 73^{10} +O\left(73^{ 11 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(5,6)$ |
| $(1,2,3)(4,6,5)$ |
| $(3,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,5)(2,4)(3,6)$ | $-3$ |
| $3$ | $2$ | $(3,6)$ | $1$ |
| $3$ | $2$ | $(1,5)(3,6)$ | $-1$ |
| $6$ | $2$ | $(1,2)(4,5)$ | $-1$ |
| $6$ | $2$ | $(1,2)(3,6)(4,5)$ | $1$ |
| $8$ | $3$ | $(1,2,3)(4,6,5)$ | $0$ |
| $6$ | $4$ | $(1,3,5,6)$ | $-1$ |
| $6$ | $4$ | $(1,4,5,2)(3,6)$ | $1$ |
| $8$ | $6$ | $(1,2,3,5,4,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
