Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 307 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 307 }$: $ x^{2} + 306 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 89 + 70\cdot 307 + 100\cdot 307^{2} + 232\cdot 307^{3} + 94\cdot 307^{4} +O\left(307^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 66 a + 59 + \left(44 a + 185\right)\cdot 307 + \left(250 a + 154\right)\cdot 307^{2} + \left(211 a + 116\right)\cdot 307^{3} + \left(240 a + 254\right)\cdot 307^{4} +O\left(307^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 241 a + 125 + \left(262 a + 163\right)\cdot 307 + \left(56 a + 53\right)\cdot 307^{2} + \left(95 a + 78\right)\cdot 307^{3} + \left(66 a + 283\right)\cdot 307^{4} +O\left(307^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 23 + 57\cdot 307 + 270\cdot 307^{2} + 287\cdot 307^{3} + 23\cdot 307^{4} +O\left(307^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 230 a + 259 + \left(16 a + 249\right)\cdot 307 + \left(19 a + 271\right)\cdot 307^{2} + \left(173 a + 292\right)\cdot 307^{3} + \left(143 a + 75\right)\cdot 307^{4} +O\left(307^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 77 a + 182 + \left(290 a + 36\right)\cdot 307 + \left(287 a + 274\right)\cdot 307^{2} + \left(133 a + 139\right)\cdot 307^{3} + \left(163 a + 46\right)\cdot 307^{4} +O\left(307^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 187 + 158\cdot 307 + 103\cdot 307^{2} + 80\cdot 307^{3} + 142\cdot 307^{4} +O\left(307^{ 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 values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$14$ |
| $21$ |
$2$ |
$(1,2)$ |
$-6$ |
| $105$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$-2$ |
| $105$ |
$2$ |
$(1,2)(3,4)$ |
$2$ |
| $70$ |
$3$ |
$(1,2,3)$ |
$2$ |
| $280$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$-1$ |
| $210$ |
$4$ |
$(1,2,3,4)$ |
$0$ |
| $630$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$0$ |
| $504$ |
$5$ |
$(1,2,3,4,5)$ |
$-1$ |
| $210$ |
$6$ |
$(1,2,3)(4,5)(6,7)$ |
$2$ |
| $420$ |
$6$ |
$(1,2,3)(4,5)$ |
$0$ |
| $840$ |
$6$ |
$(1,2,3,4,5,6)$ |
$1$ |
| $720$ |
$7$ |
$(1,2,3,4,5,6,7)$ |
$0$ |
| $504$ |
$10$ |
$(1,2,3,4,5)(6,7)$ |
$-1$ |
| $420$ |
$12$ |
$(1,2,3,4)(5,6,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
