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:
| $r_{ 1 }$ |
$=$ |
$ 30 a + 187 + \left(248 a + 80\right)\cdot 251 + \left(197 a + 89\right)\cdot 251^{2} + \left(25 a + 98\right)\cdot 251^{3} + \left(38 a + 53\right)\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 187 a + 4 + \left(245 a + 12\right)\cdot 251 + \left(181 a + 246\right)\cdot 251^{2} + \left(131 a + 14\right)\cdot 251^{3} + \left(48 a + 182\right)\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 64 a + 181 + \left(5 a + 28\right)\cdot 251 + \left(69 a + 132\right)\cdot 251^{2} + \left(119 a + 14\right)\cdot 251^{3} + \left(202 a + 236\right)\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 35 + 91\cdot 251 + 40\cdot 251^{2} + 238\cdot 251^{3} + 162\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 221 a + 206 + \left(2 a + 24\right)\cdot 251 + \left(53 a + 117\right)\cdot 251^{2} + \left(225 a + 132\right)\cdot 251^{3} + \left(212 a + 119\right)\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 28 a + 195 + \left(150 a + 223\right)\cdot 251 + \left(43 a + 193\right)\cdot 251^{2} + \left(107 a + 168\right)\cdot 251^{3} + \left(200 a + 29\right)\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 223 a + 196 + \left(100 a + 40\right)\cdot 251 + \left(207 a + 185\right)\cdot 251^{2} + \left(143 a + 85\right)\cdot 251^{3} + \left(50 a + 220\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.
