Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(52\)\(\medspace = 2^{2} \cdot 13 \) |
| Artin field: | Galois closure of 6.0.23762752.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{52}(23,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} + 13x^{4} + 26x^{2} + 13 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$:
\( x^{2} + 70x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 70 a + 41 + \left(62 a + 13\right)\cdot 73 + \left(24 a + 67\right)\cdot 73^{2} + \left(47 a + 50\right)\cdot 73^{3} + \left(15 a + 36\right)\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 63 a + 15 + \left(35 a + 14\right)\cdot 73 + \left(11 a + 37\right)\cdot 73^{2} + \left(46 a + 9\right)\cdot 73^{3} + \left(59 a + 43\right)\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 37 a + 54 + \left(46 a + 21\right)\cdot 73 + \left(59 a + 43\right)\cdot 73^{2} + \left(55 a + 55\right)\cdot 73^{3} + \left(4 a + 20\right)\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 3 a + 32 + \left(10 a + 59\right)\cdot 73 + \left(48 a + 5\right)\cdot 73^{2} + \left(25 a + 22\right)\cdot 73^{3} + \left(57 a + 36\right)\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 10 a + 58 + \left(37 a + 58\right)\cdot 73 + \left(61 a + 35\right)\cdot 73^{2} + \left(26 a + 63\right)\cdot 73^{3} + \left(13 a + 29\right)\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 36 a + 19 + \left(26 a + 51\right)\cdot 73 + \left(13 a + 29\right)\cdot 73^{2} + \left(17 a + 17\right)\cdot 73^{3} + \left(68 a + 52\right)\cdot 73^{4} +O(73^{5})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 6 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 6 }$ | Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-1$ |
| $1$ | $3$ | $(1,6,2)(3,5,4)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,2,6)(3,4,5)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,3,2,4,6,5)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,5,6,4,2,3)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.