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:
| $r_{ 1 }$ |
$=$ |
$ 45 a + 16 + \left(28 a + 36\right)\cdot 73 + \left(33 a + 12\right)\cdot 73^{2} + \left(56 a + 10\right)\cdot 73^{3} + \left(67 a + 12\right)\cdot 73^{4} + \left(44 a + 22\right)\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 28 a + 5 + \left(44 a + 4\right)\cdot 73 + \left(39 a + 11\right)\cdot 73^{2} + 16 a\cdot 73^{3} + \left(5 a + 13\right)\cdot 73^{4} + \left(28 a + 16\right)\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 69 a + 15 + 64 a\cdot 73 + \left(42 a + 55\right)\cdot 73^{2} + \left(47 a + 10\right)\cdot 73^{3} + \left(49 a + 33\right)\cdot 73^{4} + \left(a + 55\right)\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 4 a + 3 + \left(8 a + 53\right)\cdot 73 + \left(30 a + 45\right)\cdot 73^{2} + \left(25 a + 37\right)\cdot 73^{3} + \left(23 a + 61\right)\cdot 73^{4} + \left(71 a + 10\right)\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 14 + 34\cdot 73 + 56\cdot 73^{2} + 20\cdot 73^{3} + 57\cdot 73^{4} + 67\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 22 + 18\cdot 73 + 38\cdot 73^{2} + 66\cdot 73^{3} + 41\cdot 73^{4} + 46\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,3)$ |
| $(1,2)(3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $8$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $40$ | $3$ | $(1,2,3)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $72$ | $5$ | $(1,2,3,4,5)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
| $72$ | $5$ | $(1,3,4,5,2)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
