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:
| $r_{ 1 }$ |
$=$ |
$ 3 + 31\cdot 137 + 3\cdot 137^{2} + 16\cdot 137^{3} + 61\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 115 a + 71 + \left(136 a + 125\right)\cdot 137 + \left(121 a + 96\right)\cdot 137^{2} + \left(69 a + 39\right)\cdot 137^{3} + \left(26 a + 103\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 11 a + 81 + \left(121 a + 18\right)\cdot 137 + \left(26 a + 58\right)\cdot 137^{2} + \left(78 a + 63\right)\cdot 137^{3} + \left(69 a + 121\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 22 a + 76 + 9\cdot 137 + \left(15 a + 7\right)\cdot 137^{2} + \left(67 a + 63\right)\cdot 137^{3} + \left(110 a + 55\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 126 a + 10 + \left(15 a + 49\right)\cdot 137 + \left(110 a + 98\right)\cdot 137^{2} + \left(58 a + 94\right)\cdot 137^{3} + \left(67 a + 49\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 35 + 40\cdot 137 + 10\cdot 137^{2} + 134\cdot 137^{3} + 19\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$16$ |
| $15$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$0$ |
| $15$ |
$2$ |
$(1,2)$ |
$0$ |
| $45$ |
$2$ |
$(1,2)(3,4)$ |
$0$ |
| $40$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$-2$ |
| $40$ |
$3$ |
$(1,2,3)$ |
$-2$ |
| $90$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$0$ |
| $90$ |
$4$ |
$(1,2,3,4)$ |
$0$ |
| $144$ |
$5$ |
$(1,2,3,4,5)$ |
$1$ |
| $120$ |
$6$ |
$(1,2,3,4,5,6)$ |
$0$ |
| $120$ |
$6$ |
$(1,2,3)(4,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
