Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$: $ x^{2} + 70 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 a + 33 + \left(70 a + 4\right)\cdot 73 + \left(62 a + 45\right)\cdot 73^{2} + \left(59 a + 29\right)\cdot 73^{3} + \left(25 a + 36\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 24 + 44\cdot 73 + 38\cdot 73^{2} + 28\cdot 73^{3} + 22\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 48 + 50\cdot 73 + 26\cdot 73^{2} + 22\cdot 73^{3} + 41\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 36 a + 14 + \left(58 a + 68\right)\cdot 73 + \left(71 a + 39\right)\cdot 73^{2} + \left(36 a + 49\right)\cdot 73^{3} + \left(22 a + 53\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 37 a + 49 + \left(14 a + 61\right)\cdot 73 + \left(a + 50\right)\cdot 73^{2} + \left(36 a + 15\right)\cdot 73^{3} + \left(50 a + 11\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 67 a + 51 + \left(2 a + 62\right)\cdot 73 + \left(10 a + 17\right)\cdot 73^{2} + 13 a\cdot 73^{3} + \left(47 a + 54\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,3)$ |
| $(1,2)(3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$9$ |
| $45$ |
$2$ |
$(1,2)(3,4)$ |
$1$ |
| $40$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$0$ |
| $40$ |
$3$ |
$(1,2,3)$ |
$0$ |
| $90$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$1$ |
| $72$ |
$5$ |
$(1,2,3,4,5)$ |
$-1$ |
| $72$ |
$5$ |
$(1,3,4,5,2)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
