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:
| $r_{ 1 }$ |
$=$ |
$ 57 a + 47 + \left(58 a + 13\right)\cdot 73 + \left(66 a + 9\right)\cdot 73^{2} + \left(42 a + 70\right)\cdot 73^{3} + \left(49 a + 25\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 57 a + 27 + \left(58 a + 24\right)\cdot 73 + \left(66 a + 63\right)\cdot 73^{2} + \left(42 a + 8\right)\cdot 73^{3} + \left(49 a + 47\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 16 a + 61 + \left(14 a + 11\right)\cdot 73 + \left(6 a + 39\right)\cdot 73^{2} + \left(30 a + 72\right)\cdot 73^{3} + \left(23 a + 19\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 16 a + 52 + \left(14 a + 70\right)\cdot 73 + \left(6 a + 58\right)\cdot 73^{2} + \left(30 a + 70\right)\cdot 73^{3} + \left(23 a + 6\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 16 a + 72 + \left(14 a + 59\right)\cdot 73 + \left(6 a + 4\right)\cdot 73^{2} + \left(30 a + 59\right)\cdot 73^{3} + \left(23 a + 58\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 57 a + 36 + \left(58 a + 38\right)\cdot 73 + \left(66 a + 43\right)\cdot 73^{2} + \left(42 a + 10\right)\cdot 73^{3} + \left(49 a + 60\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3,2,5,6,4)$ |
| $(1,5)(2,4)(3,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,5)(2,4)(3,6)$ | $-1$ |
| $1$ | $3$ | $(1,2,6)(3,5,4)$ | $-\zeta_{3} - 1$ |
| $1$ | $3$ | $(1,6,2)(3,4,5)$ | $\zeta_{3}$ |
| $1$ | $6$ | $(1,3,2,5,6,4)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,4,6,5,2,3)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
