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:
| $r_{ 1 }$ |
$=$ |
$ 24 a + 21 + \left(69 a + 21\right)\cdot 73 + \left(56 a + 44\right)\cdot 73^{2} + \left(45 a + 62\right)\cdot 73^{3} + \left(39 a + 22\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 34 a + 23 + \left(52 a + 40\right)\cdot 73 + \left(70 a + 72\right)\cdot 73^{2} + \left(17 a + 31\right)\cdot 73^{3} + \left(6 a + 65\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 17 + 5\cdot 73 + 72\cdot 73^{2} + 50\cdot 73^{3} + 57\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 16 + 13\cdot 73 + 47\cdot 73^{2} + 4\cdot 73^{3} + 17\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 71 + 61\cdot 73 + 42\cdot 73^{2} + 56\cdot 73^{3} + 39\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 39 a + 52 + \left(20 a + 17\right)\cdot 73 + \left(2 a + 13\right)\cdot 73^{2} + \left(55 a + 15\right)\cdot 73^{3} + \left(66 a + 66\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 49 a + 20 + \left(3 a + 59\right)\cdot 73 + \left(16 a + 72\right)\cdot 73^{2} + \left(27 a + 69\right)\cdot 73^{3} + \left(33 a + 22\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 7 }$
| Cycle notation |
| $(1,2,3,4,5,6,7)$ |
| $(1,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 7 }$
| Character value |
| $1$ | $1$ | $()$ | $6$ |
| $21$ | $2$ | $(1,2)$ | $-4$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $2$ |
| $70$ | $3$ | $(1,2,3)$ | $3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $210$ | $4$ | $(1,2,3,4)$ | $-2$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $-1$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
