Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 137 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 137 }$: $ x^{2} + 131 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 5 a + 105 + \left(5 a + 70\right)\cdot 137 + \left(104 a + 67\right)\cdot 137^{2} + \left(124 a + 100\right)\cdot 137^{3} + \left(120 a + 76\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 35 a + 113 + \left(58 a + 115\right)\cdot 137 + \left(92 a + 67\right)\cdot 137^{2} + \left(74 a + 37\right)\cdot 137^{3} + \left(5 a + 49\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 102 a + 49 + \left(78 a + 19\right)\cdot 137 + \left(44 a + 16\right)\cdot 137^{2} + \left(62 a + 119\right)\cdot 137^{3} + \left(131 a + 7\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 90 + 125\cdot 137 + 103\cdot 137^{2} + 91\cdot 137^{3} + 12\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 59 + 120\cdot 137 + 16\cdot 137^{2} + 2\cdot 137^{3} + 135\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 132 a + 135 + \left(131 a + 95\right)\cdot 137 + \left(32 a + 1\right)\cdot 137^{2} + \left(12 a + 60\right)\cdot 137^{3} + \left(16 a + 129\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,3)$ |
| $(1,2)(3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $8$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $40$ | $3$ | $(1,2,3)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $72$ | $5$ | $(1,2,3,4,5)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
| $72$ | $5$ | $(1,3,4,5,2)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
