Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 251 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 251 }$: $ x^{2} + 242 x + 6 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 a + 30 + \left(86 a + 183\right)\cdot 251 + \left(118 a + 116\right)\cdot 251^{2} + \left(110 a + 229\right)\cdot 251^{3} + \left(7 a + 194\right)\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 100 a + 165 + \left(78 a + 209\right)\cdot 251 + \left(138 a + 147\right)\cdot 251^{2} + \left(212 a + 47\right)\cdot 251^{3} + \left(56 a + 216\right)\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 245 a + 84 + \left(164 a + 198\right)\cdot 251 + \left(132 a + 91\right)\cdot 251^{2} + \left(140 a + 101\right)\cdot 251^{3} + \left(243 a + 151\right)\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 63 a + 83 + \left(160 a + 112\right)\cdot 251 + \left(97 a + 169\right)\cdot 251^{2} + \left(18 a + 126\right)\cdot 251^{3} + \left(176 a + 133\right)\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 151 a + 61 + \left(172 a + 62\right)\cdot 251 + \left(112 a + 59\right)\cdot 251^{2} + \left(38 a + 65\right)\cdot 251^{3} + \left(194 a + 13\right)\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 184 + 251 + 33\cdot 251^{2} + 239\cdot 251^{3} + 100\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 188 a + 148 + \left(90 a + 236\right)\cdot 251 + \left(153 a + 134\right)\cdot 251^{2} + \left(232 a + 194\right)\cdot 251^{3} + \left(74 a + 193\right)\cdot 251^{4} +O\left(251^{ 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 |
| $1$ | $1$ | $()$ | $20$ |
| $21$ | $2$ | $(1,2)$ | $0$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $-4$ |
| $70$ | $3$ | $(1,2,3)$ | $2$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $2$ |
| $210$ | $4$ | $(1,2,3,4)$ | $0$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $2$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $-1$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $0$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
