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:
| $r_{ 1 }$ |
$=$ |
$ 31 a + 62 + \left(70 a + 36\right)\cdot 73 + \left(36 a + 3\right)\cdot 73^{2} + \left(44 a + 65\right)\cdot 73^{3} + \left(22 a + 4\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 42 a + 9 + \left(2 a + 71\right)\cdot 73 + \left(36 a + 43\right)\cdot 73^{2} + \left(28 a + 15\right)\cdot 73^{3} + \left(50 a + 28\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 27 + 50\cdot 73 + 30\cdot 73^{2} + 3\cdot 73^{3} +O\left(73^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 42 a + 71 + \left(27 a + 9\right)\cdot 73 + \left(37 a + 28\right)\cdot 73^{2} + \left(45 a + 54\right)\cdot 73^{3} + \left(48 a + 42\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 31 a + 51 + \left(45 a + 50\right)\cdot 73 + \left(35 a + 39\right)\cdot 73^{2} + \left(27 a + 7\right)\cdot 73^{3} + \left(24 a + 70\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2,3)$ |
| $(3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $20$ | $3$ | $(1,2,3)$ | $0$ |
| $12$ | $5$ | $(1,2,3,4,5)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
| $12$ | $5$ | $(1,3,4,5,2)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
