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:
| $r_{ 1 }$ |
$=$ |
$ 20 a + 14 + \left(31 a + 68\right)\cdot 73 + \left(45 a + 24\right)\cdot 73^{2} + \left(48 a + 20\right)\cdot 73^{3} + \left(37 a + 33\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 62 + 17\cdot 73 + 21\cdot 73^{2} + 71\cdot 73^{3} + 54\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 68 + 58\cdot 73 + 32\cdot 73^{2} + 39\cdot 73^{3} + 7\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 23 a + 39 + \left(38 a + 29\right)\cdot 73 + \left(57 a + 47\right)\cdot 73^{2} + \left(29 a + 40\right)\cdot 73^{3} + \left(57 a + 14\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 53 a + 1 + \left(41 a + 69\right)\cdot 73 + \left(27 a + 56\right)\cdot 73^{2} + \left(24 a + 47\right)\cdot 73^{3} + \left(35 a + 24\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 50 a + 35 + \left(34 a + 48\right)\cdot 73 + \left(15 a + 35\right)\cdot 73^{2} + \left(43 a + 72\right)\cdot 73^{3} + \left(15 a + 10\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $16$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $15$ | $2$ | $(1,2)$ | $0$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-2$ |
| $40$ | $3$ | $(1,2,3)$ | $-2$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $90$ | $4$ | $(1,2,3,4)$ | $0$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
