Basic invariants
Dimension: | $1$ |
Group: | $C_6$ |
Conductor: | \(91\)\(\medspace = 7 \cdot 13 \) |
Artin field: | Galois closure of 6.0.36924979.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_6$ |
Parity: | odd |
Dirichlet character: | \(\chi_{91}(38,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - x^{5} + 22x^{4} - 22x^{3} + 148x^{2} - 148x + 337 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 71 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 71 }$: \( x^{2} + 69x + 7 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 63 a + 53 + \left(39 a + 56\right)\cdot 71 + \left(12 a + 68\right)\cdot 71^{2} + \left(35 a + 49\right)\cdot 71^{3} + \left(31 a + 44\right)\cdot 71^{4} +O(71^{5})\)
$r_{ 2 }$ |
$=$ |
\( 61 a + 8 + \left(a + 41\right)\cdot 71 + \left(54 a + 61\right)\cdot 71^{2} + \left(57 a + 44\right)\cdot 71^{3} + \left(65 a + 51\right)\cdot 71^{4} +O(71^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 10 a + 59 + \left(69 a + 54\right)\cdot 71 + \left(16 a + 25\right)\cdot 71^{2} + \left(13 a + 35\right)\cdot 71^{3} + \left(5 a + 54\right)\cdot 71^{4} +O(71^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 40 a + 24 + \left(26 a + 22\right)\cdot 71 + \left(25 a + 60\right)\cdot 71^{2} + \left(65 a + 5\right)\cdot 71^{3} + \left(7 a + 55\right)\cdot 71^{4} +O(71^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 8 a + 37 + \left(31 a + 2\right)\cdot 71 + \left(58 a + 54\right)\cdot 71^{2} + \left(35 a + 36\right)\cdot 71^{3} + \left(39 a + 1\right)\cdot 71^{4} +O(71^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 31 a + 33 + \left(44 a + 35\right)\cdot 71 + \left(45 a + 13\right)\cdot 71^{2} + \left(5 a + 40\right)\cdot 71^{3} + \left(63 a + 5\right)\cdot 71^{4} +O(71^{5})\)
| |
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 |
$1$ | $1$ | $()$ | $1$ |
$1$ | $2$ | $(1,5)(2,3)(4,6)$ | $-1$ |
$1$ | $3$ | $(1,3,4)(2,6,5)$ | $\zeta_{3}$ |
$1$ | $3$ | $(1,4,3)(2,5,6)$ | $-\zeta_{3} - 1$ |
$1$ | $6$ | $(1,6,3,5,4,2)$ | $\zeta_{3} + 1$ |
$1$ | $6$ | $(1,2,4,5,3,6)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.