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:
| $r_{ 1 }$ |
$=$ |
$ 61 a + 28 + \left(14 a + 25\right)\cdot 73 + \left(57 a + 30\right)\cdot 73^{2} + \left(34 a + 65\right)\cdot 73^{3} + \left(15 a + 71\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 60 + 46\cdot 73 + 62\cdot 73^{2} + 17\cdot 73^{3} + 22\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 12 a + 65 + \left(58 a + 8\right)\cdot 73 + \left(15 a + 41\right)\cdot 73^{2} + \left(38 a + 39\right)\cdot 73^{3} + \left(57 a + 10\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 61 + 21\cdot 73 + 68\cdot 73^{2} + 67\cdot 73^{3} + 71\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 72 a + 42 + \left(41 a + 67\right)\cdot 73 + \left(27 a + 60\right)\cdot 73^{2} + \left(44 a + 70\right)\cdot 73^{3} + \left(63 a + 20\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ a + 39 + \left(31 a + 48\right)\cdot 73 + \left(45 a + 28\right)\cdot 73^{2} + \left(28 a + 30\right)\cdot 73^{3} + \left(9 a + 21\right)\cdot 73^{4} +O\left(73^{ 5 }\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 |
| $1$ | $1$ | $()$ | $8$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $40$ | $3$ | $(1,2,3)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $72$ | $5$ | $(1,2,3,4,5)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
| $72$ | $5$ | $(1,3,4,5,2)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.
