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:
| $r_{ 1 }$ |
$=$ |
$ 10 + 60\cdot 73 + 52\cdot 73^{2} + 34\cdot 73^{3} + 70\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 34 + 33\cdot 73 + 71\cdot 73^{2} + 7\cdot 73^{3} + 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 38 a + 66 + \left(63 a + 43\right)\cdot 73 + \left(49 a + 24\right)\cdot 73^{2} + \left(45 a + 60\right)\cdot 73^{3} + \left(54 a + 49\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 53 + 64\cdot 73 + 4\cdot 73^{2} + 73^{3} + 56\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 35 a + 34 + \left(9 a + 50\right)\cdot 73 + \left(23 a + 37\right)\cdot 73^{2} + \left(27 a + 1\right)\cdot 73^{3} + \left(18 a + 22\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 33 a + 35 + \left(33 a + 22\right)\cdot 73 + \left(59 a + 14\right)\cdot 73^{2} + \left(53 a + 42\right)\cdot 73^{3} + \left(56 a + 24\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 40 a + 61 + \left(39 a + 16\right)\cdot 73 + \left(13 a + 13\right)\cdot 73^{2} + \left(19 a + 71\right)\cdot 73^{3} + \left(16 a + 67\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 values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$14$ |
| $21$ |
$2$ |
$(1,2)$ |
$4$ |
| $105$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$0$ |
| $105$ |
$2$ |
$(1,2)(3,4)$ |
$2$ |
| $70$ |
$3$ |
$(1,2,3)$ |
$-1$ |
| $280$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$2$ |
| $210$ |
$4$ |
$(1,2,3,4)$ |
$-2$ |
| $630$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$0$ |
| $504$ |
$5$ |
$(1,2,3,4,5)$ |
$-1$ |
| $210$ |
$6$ |
$(1,2,3)(4,5)(6,7)$ |
$-1$ |
| $420$ |
$6$ |
$(1,2,3)(4,5)$ |
$1$ |
| $840$ |
$6$ |
$(1,2,3,4,5,6)$ |
$0$ |
| $720$ |
$7$ |
$(1,2,3,4,5,6,7)$ |
$0$ |
| $504$ |
$10$ |
$(1,2,3,4,5)(6,7)$ |
$-1$ |
| $420$ |
$12$ |
$(1,2,3,4)(5,6,7)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
