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:
| $r_{ 1 }$ |
$=$ |
$ 70 a + 41 + \left(62 a + 13\right)\cdot 73 + \left(24 a + 67\right)\cdot 73^{2} + \left(47 a + 50\right)\cdot 73^{3} + \left(15 a + 36\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 63 a + 15 + \left(35 a + 14\right)\cdot 73 + \left(11 a + 37\right)\cdot 73^{2} + \left(46 a + 9\right)\cdot 73^{3} + \left(59 a + 43\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 37 a + 54 + \left(46 a + 21\right)\cdot 73 + \left(59 a + 43\right)\cdot 73^{2} + \left(55 a + 55\right)\cdot 73^{3} + \left(4 a + 20\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 3 a + 32 + \left(10 a + 59\right)\cdot 73 + \left(48 a + 5\right)\cdot 73^{2} + \left(25 a + 22\right)\cdot 73^{3} + \left(57 a + 36\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 10 a + 58 + \left(37 a + 58\right)\cdot 73 + \left(61 a + 35\right)\cdot 73^{2} + \left(26 a + 63\right)\cdot 73^{3} + \left(13 a + 29\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 36 a + 19 + \left(26 a + 51\right)\cdot 73 + \left(13 a + 29\right)\cdot 73^{2} + \left(17 a + 17\right)\cdot 73^{3} + \left(68 a + 52\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)(2,5)(3,6)$ |
| $(1,6,2)(3,5,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-1$ |
| $1$ | $3$ | $(1,6,2)(3,5,4)$ | $-\zeta_{3} - 1$ |
| $1$ | $3$ | $(1,2,6)(3,4,5)$ | $\zeta_{3}$ |
| $1$ | $6$ | $(1,3,2,4,6,5)$ | $\zeta_{3} + 1$ |
| $1$ | $6$ | $(1,5,6,4,2,3)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
