Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 151 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 151 }$: $ x^{2} + 149 x + 6 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 142 a + 8 + \left(17 a + 84\right)\cdot 151 + \left(49 a + 63\right)\cdot 151^{2} + \left(61 a + 45\right)\cdot 151^{3} + \left(140 a + 126\right)\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 10 + 138\cdot 151 + 38\cdot 151^{2} + 62\cdot 151^{3} + 91\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 95 a + 133 + \left(7 a + 97\right)\cdot 151 + \left(14 a + 53\right)\cdot 151^{2} + \left(93 a + 149\right)\cdot 151^{3} + \left(44 a + 107\right)\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 128 a + 93 + \left(143 a + 139\right)\cdot 151 + \left(44 a + 141\right)\cdot 151^{2} + \left(91 a + 110\right)\cdot 151^{3} + \left(35 a + 150\right)\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 56 a + 21 + \left(143 a + 18\right)\cdot 151 + \left(136 a + 74\right)\cdot 151^{2} + \left(57 a + 19\right)\cdot 151^{3} + \left(106 a + 104\right)\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 9 a + 141 + \left(133 a + 128\right)\cdot 151 + \left(101 a + 143\right)\cdot 151^{2} + \left(89 a + 118\right)\cdot 151^{3} + \left(10 a + 43\right)\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 23 a + 47 + \left(7 a + 148\right)\cdot 151 + \left(106 a + 87\right)\cdot 151^{2} + \left(59 a + 97\right)\cdot 151^{3} + \left(115 a + 130\right)\cdot 151^{4} +O\left(151^{ 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 |
| $1$ | $1$ | $()$ | $14$ |
| $21$ | $2$ | $(1,2)$ | $6$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $2$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $2$ |
| $70$ | $3$ | $(1,2,3)$ | $2$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $210$ | $4$ | $(1,2,3,4)$ | $0$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $2$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $-1$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
