Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 349 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 349 }$: $ x^{2} + 348 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 210 a + 18 + \left(44 a + 81\right)\cdot 349 + \left(10 a + 27\right)\cdot 349^{2} + \left(244 a + 282\right)\cdot 349^{3} + \left(213 a + 210\right)\cdot 349^{4} +O\left(349^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 139 a + 228 + \left(304 a + 264\right)\cdot 349 + \left(338 a + 341\right)\cdot 349^{2} + \left(104 a + 166\right)\cdot 349^{3} + \left(135 a + 180\right)\cdot 349^{4} +O\left(349^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 81 + 311\cdot 349 + 255\cdot 349^{2} + 27\cdot 349^{3} + 40\cdot 349^{4} +O\left(349^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 223 a + 65 + \left(240 a + 108\right)\cdot 349 + \left(35 a + 95\right)\cdot 349^{2} + \left(291 a + 40\right)\cdot 349^{3} + \left(208 a + 87\right)\cdot 349^{4} +O\left(349^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 126 a + 288 + \left(108 a + 125\right)\cdot 349 + \left(313 a + 239\right)\cdot 349^{2} + \left(57 a + 295\right)\cdot 349^{3} + \left(140 a + 4\right)\cdot 349^{4} +O\left(349^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 266 + 295\cdot 349 + 145\cdot 349^{2} + 53\cdot 349^{3} + 125\cdot 349^{4} +O\left(349^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 101 + 209\cdot 349 + 290\cdot 349^{2} + 180\cdot 349^{3} + 49\cdot 349^{4} +O\left(349^{ 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.
