Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$: $ x^{2} + 70 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 60 a + 50 + \left(18 a + 23\right)\cdot 73 + \left(20 a + 32\right)\cdot 73^{2} + \left(10 a + 44\right)\cdot 73^{3} + \left(29 a + 28\right)\cdot 73^{4} + \left(53 a + 5\right)\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 25 a + 18 + \left(68 a + 13\right)\cdot 73 + \left(69 a + 58\right)\cdot 73^{2} + \left(32 a + 20\right)\cdot 73^{3} + \left(46 a + 4\right)\cdot 73^{4} + \left(24 a + 9\right)\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 39 a + 2 + \left(62 a + 56\right)\cdot 73 + \left(17 a + 4\right)\cdot 73^{2} + \left(58 a + 19\right)\cdot 73^{3} + \left(27 a + 45\right)\cdot 73^{4} + \left(13 a + 9\right)\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 13 a + 11 + \left(54 a + 20\right)\cdot 73 + \left(52 a + 1\right)\cdot 73^{2} + \left(62 a + 55\right)\cdot 73^{3} + \left(43 a + 32\right)\cdot 73^{4} + \left(19 a + 63\right)\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 48 a + 20 + \left(4 a + 47\right)\cdot 73 + \left(3 a + 53\right)\cdot 73^{2} + \left(40 a + 49\right)\cdot 73^{3} + \left(26 a + 37\right)\cdot 73^{4} + \left(48 a + 36\right)\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 34 a + 46 + \left(10 a + 58\right)\cdot 73 + \left(55 a + 68\right)\cdot 73^{2} + \left(14 a + 29\right)\cdot 73^{3} + \left(45 a + 70\right)\cdot 73^{4} + \left(59 a + 21\right)\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,5,3)(2,4,6)$ |
| $(1,4,3,6,5,2)$ |
| $(2,6,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,6)(2,3)(4,5)$ | $0$ |
| $1$ | $3$ | $(1,3,5)(2,4,6)$ | $-2 \zeta_{3} - 2$ |
| $1$ | $3$ | $(1,5,3)(2,6,4)$ | $2 \zeta_{3}$ |
| $2$ | $3$ | $(1,5,3)(2,4,6)$ | $-1$ |
| $2$ | $3$ | $(2,6,4)$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(2,4,6)$ | $-\zeta_{3}$ |
| $3$ | $6$ | $(1,4,3,6,5,2)$ | $0$ |
| $3$ | $6$ | $(1,2,5,6,3,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
