Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 139 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 139 }$: $ x^{2} + 138 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 59 a + 67 + \left(79 a + 78\right)\cdot 139 + \left(117 a + 23\right)\cdot 139^{2} + \left(53 a + 126\right)\cdot 139^{3} + \left(15 a + 101\right)\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 129 a + 91 + \left(96 a + 120\right)\cdot 139 + \left(87 a + 17\right)\cdot 139^{2} + \left(73 a + 5\right)\cdot 139^{3} + \left(78 a + 49\right)\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 79 + 51\cdot 139 + 101\cdot 139^{2} + 75\cdot 139^{3} + 137\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 80 a + 126 + \left(59 a + 98\right)\cdot 139 + \left(21 a + 61\right)\cdot 139^{2} + \left(85 a + 62\right)\cdot 139^{3} + \left(123 a + 63\right)\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 74 + 18\cdot 139 + 71\cdot 139^{2} + 56\cdot 139^{3} + 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 40 + 99\cdot 139 + 132\cdot 139^{2} + 99\cdot 139^{3} + 9\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 10 a + 81 + \left(42 a + 88\right)\cdot 139 + \left(51 a + 8\right)\cdot 139^{2} + \left(65 a + 130\right)\cdot 139^{3} + \left(60 a + 53\right)\cdot 139^{4} +O\left(139^{ 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 |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$6$ |
| $21$ |
$2$ |
$(1,2)$ |
$4$ |
| $105$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$0$ |
| $105$ |
$2$ |
$(1,2)(3,4)$ |
$2$ |
| $70$ |
$3$ |
$(1,2,3)$ |
$3$ |
| $280$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$0$ |
| $210$ |
$4$ |
$(1,2,3,4)$ |
$2$ |
| $630$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$0$ |
| $504$ |
$5$ |
$(1,2,3,4,5)$ |
$1$ |
| $210$ |
$6$ |
$(1,2,3)(4,5)(6,7)$ |
$-1$ |
| $420$ |
$6$ |
$(1,2,3)(4,5)$ |
$1$ |
| $840$ |
$6$ |
$(1,2,3,4,5,6)$ |
$0$ |
| $720$ |
$7$ |
$(1,2,3,4,5,6,7)$ |
$-1$ |
| $504$ |
$10$ |
$(1,2,3,4,5)(6,7)$ |
$-1$ |
| $420$ |
$12$ |
$(1,2,3,4)(5,6,7)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
