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:
| $r_{ 1 }$ |
$=$ |
$ 3 a + 42 + \left(61 a + 59\right)\cdot 73 + \left(58 a + 46\right)\cdot 73^{2} + \left(27 a + 16\right)\cdot 73^{3} + \left(31 a + 23\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 9 a + 64 + \left(64 a + 52\right)\cdot 73 + \left(37 a + 16\right)\cdot 73^{2} + \left(24 a + 70\right)\cdot 73^{3} + \left(67 a + 25\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 64 a + 18 + \left(8 a + 17\right)\cdot 73 + \left(35 a + 66\right)\cdot 73^{2} + \left(48 a + 32\right)\cdot 73^{3} + \left(5 a + 57\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 54 + 65\cdot 73 + 9\cdot 73^{2} + 15\cdot 73^{3} + 33\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 70 a + 51 + \left(11 a + 20\right)\cdot 73 + \left(14 a + 16\right)\cdot 73^{2} + \left(45 a + 41\right)\cdot 73^{3} + \left(41 a + 16\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 65 + 2\cdot 73 + 63\cdot 73^{2} + 42\cdot 73^{3} + 62\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)$ |
| $(2,3)$ |
| $(2,3,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $6$ | $2$ | $(2,3)$ | $-2$ |
| $9$ | $2$ | $(1,4)(2,3)$ | $0$ |
| $4$ | $3$ | $(2,3,6)$ | $1$ |
| $4$ | $3$ | $(1,4,5)(2,3,6)$ | $-2$ |
| $18$ | $4$ | $(1,2,4,3)(5,6)$ | $0$ |
| $12$ | $6$ | $(1,2,4,3,5,6)$ | $0$ |
| $12$ | $6$ | $(1,4,5)(2,3)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
