Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$: $ x^{2} + 70 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 41 + 40\cdot 73 + 49\cdot 73^{2} + 16\cdot 73^{3} + 41\cdot 73^{4} + 68\cdot 73^{5} + 40\cdot 73^{6} +O\left(73^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 64 a + 44 + \left(7 a + 55\right)\cdot 73 + \left(14 a + 26\right)\cdot 73^{2} + \left(53 a + 62\right)\cdot 73^{3} + \left(9 a + 13\right)\cdot 73^{4} + \left(52 a + 59\right)\cdot 73^{5} + \left(37 a + 26\right)\cdot 73^{6} +O\left(73^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 9 a + 30 + \left(65 a + 17\right)\cdot 73 + \left(58 a + 46\right)\cdot 73^{2} + \left(19 a + 10\right)\cdot 73^{3} + \left(63 a + 59\right)\cdot 73^{4} + \left(20 a + 13\right)\cdot 73^{5} + \left(35 a + 46\right)\cdot 73^{6} +O\left(73^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 33 + 32\cdot 73 + 23\cdot 73^{2} + 56\cdot 73^{3} + 31\cdot 73^{4} + 4\cdot 73^{5} + 32\cdot 73^{6} +O\left(73^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 64 a + 57 + \left(7 a + 57\right)\cdot 73 + \left(14 a + 11\right)\cdot 73^{2} + \left(53 a + 11\right)\cdot 73^{3} + \left(9 a + 10\right)\cdot 73^{4} + \left(52 a + 13\right)\cdot 73^{5} + \left(37 a + 58\right)\cdot 73^{6} +O\left(73^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 9 a + 17 + \left(65 a + 15\right)\cdot 73 + \left(58 a + 61\right)\cdot 73^{2} + \left(19 a + 61\right)\cdot 73^{3} + \left(63 a + 62\right)\cdot 73^{4} + \left(20 a + 59\right)\cdot 73^{5} + \left(35 a + 14\right)\cdot 73^{6} +O\left(73^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)$ |
| $(1,2,5)(3,6,4)$ |
| $(1,2)(3,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,4)(2,3)(5,6)$ | $-3$ |
| $3$ | $2$ | $(1,4)(2,3)$ | $-1$ |
| $3$ | $2$ | $(1,4)$ | $1$ |
| $6$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $6$ | $2$ | $(1,5)(2,3)(4,6)$ | $1$ |
| $8$ | $3$ | $(1,2,5)(3,6,4)$ | $0$ |
| $6$ | $4$ | $(1,2,4,3)$ | $-1$ |
| $6$ | $4$ | $(1,6,4,5)(2,3)$ | $1$ |
| $8$ | $6$ | $(1,3,6,4,2,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
