Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$: $ x^{2} + 70 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 57 a + 7 + \left(58 a + 47\right)\cdot 73 + 61\cdot 73^{2} + \left(14 a + 24\right)\cdot 73^{3} + \left(19 a + 38\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 16 a + 32 + \left(14 a + 20\right)\cdot 73 + \left(72 a + 5\right)\cdot 73^{2} + \left(58 a + 66\right)\cdot 73^{3} + \left(53 a + 8\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 70 + 73 + 42\cdot 73^{2} + 33\cdot 73^{3} + 11\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 23 + 8\cdot 73 + 18\cdot 73^{2} + 11\cdot 73^{3} + 15\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 68 a + 16 + \left(18 a + 3\right)\cdot 73 + \left(56 a + 44\right)\cdot 73^{2} + \left(30 a + 23\right)\cdot 73^{3} + \left(59 a + 35\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 5 a + 1 + \left(54 a + 65\right)\cdot 73 + \left(16 a + 47\right)\cdot 73^{2} + \left(42 a + 59\right)\cdot 73^{3} + \left(13 a + 36\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,4)$ |
| $(1,3)(2,5)(4,6)$ |
| $(1,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,3)(2,5)(4,6)$ | $0$ |
| $6$ | $2$ | $(1,2)$ | $2$ |
| $9$ | $2$ | $(1,2)(3,5)$ | $0$ |
| $4$ | $3$ | $(1,2,4)(3,5,6)$ | $-2$ |
| $4$ | $3$ | $(3,5,6)$ | $1$ |
| $18$ | $4$ | $(1,5,2,3)(4,6)$ | $0$ |
| $12$ | $6$ | $(1,3,2,5,4,6)$ | $0$ |
| $12$ | $6$ | $(1,2)(3,5,6)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
