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:
| $r_{ 1 }$ |
$=$ |
$ 7 a + 42 + \left(31 a + 26\right)\cdot 73 + \left(36 a + 43\right)\cdot 73^{2} + \left(9 a + 17\right)\cdot 73^{3} + \left(62 a + 56\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ a + 24 + \left(56 a + 17\right)\cdot 73 + \left(49 a + 2\right)\cdot 73^{2} + \left(44 a + 8\right)\cdot 73^{3} + \left(10 a + 46\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 72 a + 27 + \left(16 a + 38\right)\cdot 73 + \left(23 a + 22\right)\cdot 73^{2} + \left(28 a + 19\right)\cdot 73^{3} + \left(62 a + 33\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 23 + 17\cdot 73 + 48\cdot 73^{2} + 45\cdot 73^{3} + 66\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 42 + 6\cdot 73 + 54\cdot 73^{2} + 45\cdot 73^{3} + 2\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 66 a + 63 + \left(41 a + 39\right)\cdot 73 + \left(36 a + 48\right)\cdot 73^{2} + \left(63 a + 9\right)\cdot 73^{3} + \left(10 a + 14\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)$ |
| $(2,3,4)$ |
| $(2,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,5)(4,6)$ | $2$ |
| $6$ | $2$ | $(3,4)$ | $0$ |
| $9$ | $2$ | $(3,4)(5,6)$ | $0$ |
| $4$ | $3$ | $(1,5,6)(2,3,4)$ | $1$ |
| $4$ | $3$ | $(1,5,6)$ | $-2$ |
| $18$ | $4$ | $(1,2)(3,6,4,5)$ | $0$ |
| $12$ | $6$ | $(1,3,5,4,6,2)$ | $-1$ |
| $12$ | $6$ | $(1,5,6)(3,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
