Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 131 }$ to precision 11.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 131 }$: $ x^{2} + 127 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 128 + 69\cdot 131 + 18\cdot 131^{2} + 129\cdot 131^{3} + 64\cdot 131^{4} + 126\cdot 131^{5} + 9\cdot 131^{6} + 75\cdot 131^{7} + 29\cdot 131^{8} + 20\cdot 131^{9} + 92\cdot 131^{10} +O\left(131^{ 11 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 50 a + 59 + \left(5 a + 124\right)\cdot 131 + \left(71 a + 22\right)\cdot 131^{2} + \left(59 a + 23\right)\cdot 131^{3} + \left(3 a + 83\right)\cdot 131^{4} + \left(74 a + 29\right)\cdot 131^{5} + \left(92 a + 17\right)\cdot 131^{6} + \left(100 a + 12\right)\cdot 131^{7} + \left(117 a + 73\right)\cdot 131^{8} + \left(118 a + 81\right)\cdot 131^{9} + \left(114 a + 32\right)\cdot 131^{10} +O\left(131^{ 11 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 12 a + 33 + \left(43 a + 74\right)\cdot 131 + \left(102 a + 124\right)\cdot 131^{2} + \left(99 a + 25\right)\cdot 131^{3} + \left(57 a + 72\right)\cdot 131^{4} + \left(52 a + 103\right)\cdot 131^{5} + \left(a + 42\right)\cdot 131^{6} + \left(44 a + 15\right)\cdot 131^{7} + \left(66 a + 17\right)\cdot 131^{8} + \left(58 a + 86\right)\cdot 131^{9} + \left(56 a + 112\right)\cdot 131^{10} +O\left(131^{ 11 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 81 a + 128 + \left(125 a + 95\right)\cdot 131 + \left(59 a + 39\right)\cdot 131^{2} + \left(71 a + 59\right)\cdot 131^{3} + \left(127 a + 37\right)\cdot 131^{4} + \left(56 a + 60\right)\cdot 131^{5} + \left(38 a + 51\right)\cdot 131^{6} + \left(30 a + 60\right)\cdot 131^{7} + \left(13 a + 50\right)\cdot 131^{8} + \left(12 a + 46\right)\cdot 131^{9} + \left(16 a + 111\right)\cdot 131^{10} +O\left(131^{ 11 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 97 + 55\cdot 131 + 89\cdot 131^{2} + 94\cdot 131^{3} + 62\cdot 131^{4} + 79\cdot 131^{5} + 13\cdot 131^{6} + 40\cdot 131^{7} + 115\cdot 131^{8} + 35\cdot 131^{9} + 26\cdot 131^{10} +O\left(131^{ 11 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 119 a + 81 + \left(87 a + 103\right)\cdot 131 + \left(28 a + 97\right)\cdot 131^{2} + \left(31 a + 60\right)\cdot 131^{3} + \left(73 a + 72\right)\cdot 131^{4} + \left(78 a + 124\right)\cdot 131^{5} + \left(129 a + 126\right)\cdot 131^{6} + \left(86 a + 58\right)\cdot 131^{7} + \left(64 a + 107\right)\cdot 131^{8} + \left(72 a + 122\right)\cdot 131^{9} + \left(74 a + 17\right)\cdot 131^{10} +O\left(131^{ 11 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(4,5)$ |
| $(3,4)$ |
| $(1,2,3)(4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,5)(2,6)(3,4)$ | $-3$ |
| $3$ | $2$ | $(3,4)$ | $1$ |
| $3$ | $2$ | $(1,5)(3,4)$ | $-1$ |
| $6$ | $2$ | $(1,2)(5,6)$ | $-1$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $1$ |
| $8$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $6$ | $4$ | $(1,3,5,4)$ | $-1$ |
| $6$ | $4$ | $(1,6,5,2)(3,4)$ | $1$ |
| $8$ | $6$ | $(1,2,3,5,6,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
