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:
| $r_{ 1 }$ |
$=$ |
$ 151 a + 29 + \left(204 a + 186\right)\cdot 251 + \left(33 a + 43\right)\cdot 251^{2} + \left(43 a + 66\right)\cdot 251^{3} + \left(127 a + 169\right)\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 239 + 165\cdot 251 + 223\cdot 251^{2} + 181\cdot 251^{3} + 187\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 247 a + 57 + \left(15 a + 75\right)\cdot 251 + \left(104 a + 24\right)\cdot 251^{2} + \left(81 a + 75\right)\cdot 251^{3} + \left(110 a + 186\right)\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 132 + 145\cdot 251 + 242\cdot 251^{2} + 242\cdot 251^{3} + 26\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 100 a + 133 + \left(46 a + 119\right)\cdot 251 + \left(217 a + 143\right)\cdot 251^{2} + \left(207 a + 169\right)\cdot 251^{3} + \left(123 a + 15\right)\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 4 a + 21 + \left(235 a + 223\right)\cdot 251 + \left(146 a + 191\right)\cdot 251^{2} + \left(169 a + 201\right)\cdot 251^{3} + \left(140 a + 93\right)\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 142 + 88\cdot 251 + 134\cdot 251^{2} + 66\cdot 251^{3} + 73\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 |
| $1$ | $1$ | $()$ | $15$ |
| $21$ | $2$ | $(1,2)$ | $-5$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $3$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $70$ | $3$ | $(1,2,3)$ | $3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $210$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $1$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $0$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
