Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$: $ x^{2} + 70 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 47 a + 65 + \left(21 a + 43\right)\cdot 73 + \left(39 a + 12\right)\cdot 73^{2} + \left(38 a + 70\right)\cdot 73^{3} + \left(30 a + 28\right)\cdot 73^{4} + \left(45 a + 60\right)\cdot 73^{5} + \left(58 a + 30\right)\cdot 73^{6} + \left(57 a + 11\right)\cdot 73^{7} + \left(45 a + 2\right)\cdot 73^{8} +O\left(73^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 26 a + 60 + \left(51 a + 61\right)\cdot 73 + \left(33 a + 35\right)\cdot 73^{2} + 34 a\cdot 73^{3} + \left(42 a + 9\right)\cdot 73^{4} + \left(27 a + 20\right)\cdot 73^{5} + \left(14 a + 15\right)\cdot 73^{6} + \left(15 a + 53\right)\cdot 73^{7} + \left(27 a + 8\right)\cdot 73^{8} +O\left(73^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 67 a + 66 + \left(6 a + 12\right)\cdot 73 + \left(28 a + 42\right)\cdot 73^{2} + \left(38 a + 5\right)\cdot 73^{3} + \left(30 a + 61\right)\cdot 73^{4} + \left(42 a + 14\right)\cdot 73^{5} + \left(72 a + 44\right)\cdot 73^{6} + \left(56 a + 39\right)\cdot 73^{7} + \left(13 a + 8\right)\cdot 73^{8} +O\left(73^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 33 + 20\cdot 73 + 57\cdot 73^{2} + 47\cdot 73^{3} + 43\cdot 73^{4} + 19\cdot 73^{5} + 28\cdot 73^{6} + 41\cdot 73^{7} + 71\cdot 73^{8} +O\left(73^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 22 + 40\cdot 73 + 24\cdot 73^{2} + 2\cdot 73^{3} + 35\cdot 73^{4} + 65\cdot 73^{5} + 26\cdot 73^{6} + 8\cdot 73^{7} + 62\cdot 73^{8} +O\left(73^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 6 a + 48 + \left(66 a + 39\right)\cdot 73 + \left(44 a + 46\right)\cdot 73^{2} + \left(34 a + 19\right)\cdot 73^{3} + \left(42 a + 41\right)\cdot 73^{4} + \left(30 a + 38\right)\cdot 73^{5} + \left(16 a + 65\right)\cdot 73^{7} + \left(59 a + 65\right)\cdot 73^{8} +O\left(73^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,6)$ |
| $(1,3)(2,6)(4,5)$ |
| $(2,5)(4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,6)(4,5)$ | $-2$ |
| $3$ | $2$ | $(1,2)(3,6)$ | $0$ |
| $3$ | $2$ | $(1,6)(2,3)(4,5)$ | $0$ |
| $2$ | $3$ | $(1,5,2)(3,4,6)$ | $-1$ |
| $2$ | $6$ | $(1,4,2,3,5,6)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
