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:
| $r_{ 1 }$ |
$=$ |
$ 25 a + 34 + \left(63 a + 70\right)\cdot 73 + \left(38 a + 47\right)\cdot 73^{2} + \left(26 a + 26\right)\cdot 73^{3} + \left(40 a + 52\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 48 a + 36 + \left(9 a + 16\right)\cdot 73 + \left(34 a + 28\right)\cdot 73^{2} + \left(46 a + 67\right)\cdot 73^{3} + 32 a\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 26 + 63\cdot 73 + 50\cdot 73^{3} + 25\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 49 a + 25 + 14 a\cdot 73 + \left(21 a + 10\right)\cdot 73^{2} + \left(63 a + 26\right)\cdot 73^{3} + \left(a + 62\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 24 a + 26 + \left(58 a + 68\right)\cdot 73 + \left(51 a + 58\right)\cdot 73^{2} + \left(9 a + 48\right)\cdot 73^{3} + \left(71 a + 4\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $10$ | $2$ | $(1,2)$ | $2$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $20$ | $3$ | $(1,2,3)$ | $1$ |
| $30$ | $4$ | $(1,2,3,4)$ | $0$ |
| $24$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
