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:
| $r_{ 1 }$ |
$=$ |
$ 11 a + 58 + \left(16 a + 115\right)\cdot 139 + \left(130 a + 109\right)\cdot 139^{2} + \left(115 a + 14\right)\cdot 139^{3} + \left(18 a + 125\right)\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 32 + 86\cdot 139 + 65\cdot 139^{2} + 86\cdot 139^{3} + 118\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 116 a + 122 + \left(123 a + 103\right)\cdot 139 + \left(62 a + 55\right)\cdot 139^{2} + \left(14 a + 86\right)\cdot 139^{3} + \left(35 a + 36\right)\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 23 a + 99 + \left(15 a + 111\right)\cdot 139 + \left(76 a + 133\right)\cdot 139^{2} + \left(124 a + 37\right)\cdot 139^{3} + \left(103 a + 57\right)\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 128 a + 69 + \left(122 a + 120\right)\cdot 139 + \left(8 a + 84\right)\cdot 139^{2} + 23 a\cdot 139^{3} + \left(120 a + 28\right)\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 63 a + 58 + \left(62 a + 9\right)\cdot 139 + \left(92 a + 38\right)\cdot 139^{2} + \left(121 a + 11\right)\cdot 139^{3} + \left(9 a + 12\right)\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 76 a + 121 + \left(76 a + 8\right)\cdot 139 + \left(46 a + 68\right)\cdot 139^{2} + \left(17 a + 40\right)\cdot 139^{3} + \left(129 a + 39\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 value |
| $1$ | $1$ | $()$ | $15$ |
| $21$ | $2$ | $(1,2)$ | $5$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $-3$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $70$ | $3$ | $(1,2,3)$ | $3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $210$ | $4$ | $(1,2,3,4)$ | $1$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $1$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $0$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
