Basic invariants
Dimension: | $1$ |
Group: | $C_6$ |
Conductor: | \(21\)\(\medspace = 3 \cdot 7 \) |
Artin number field: | Galois closure of \(\Q(\zeta_{21})^+\) |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_6$ |
Parity: | even |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$:
\( x^{2} + 12x + 2 \)
Roots:
$r_{ 1 }$ | $=$ | \( 11 a + 9 + \left(5 a + 3\right)\cdot 13 + \left(9 a + 7\right)\cdot 13^{2} + \left(3 a + 5\right)\cdot 13^{3} + \left(2 a + 6\right)\cdot 13^{4} +O(13^{5})\) |
$r_{ 2 }$ | $=$ | \( 10 a + 4 + \left(9 a + 5\right)\cdot 13 + 12 a\cdot 13^{2} + 7 a\cdot 13^{3} + \left(12 a + 4\right)\cdot 13^{4} +O(13^{5})\) |
$r_{ 3 }$ | $=$ | \( 2 a + 7 + \left(7 a + 11\right)\cdot 13 + \left(3 a + 10\right)\cdot 13^{2} + \left(9 a + 12\right)\cdot 13^{3} + \left(10 a + 4\right)\cdot 13^{4} +O(13^{5})\) |
$r_{ 4 }$ | $=$ | \( 6 a + \left(12 a + 10\right)\cdot 13 + \left(8 a + 3\right)\cdot 13^{2} + \left(2 a + 9\right)\cdot 13^{3} + \left(9 a + 10\right)\cdot 13^{4} +O(13^{5})\) |
$r_{ 5 }$ | $=$ | \( 7 a + 6 + 3\cdot 13 + 4 a\cdot 13^{2} + \left(10 a + 3\right)\cdot 13^{3} + \left(3 a + 4\right)\cdot 13^{4} +O(13^{5})\) |
$r_{ 6 }$ | $=$ | \( 3 a + 1 + \left(3 a + 5\right)\cdot 13 + 3\cdot 13^{2} + \left(5 a + 8\right)\cdot 13^{3} + 8\cdot 13^{4} +O(13^{5})\) |
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 values | |
$c1$ | $c2$ | |||
$1$ | $1$ | $()$ | $1$ | $1$ |
$1$ | $2$ | $(1,3)(2,6)(4,5)$ | $-1$ | $-1$ |
$1$ | $3$ | $(1,4,6)(2,3,5)$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ |
$1$ | $3$ | $(1,6,4)(2,5,3)$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ |
$1$ | $6$ | $(1,2,4,3,6,5)$ | $\zeta_{3} + 1$ | $-\zeta_{3}$ |
$1$ | $6$ | $(1,5,6,3,4,2)$ | $-\zeta_{3}$ | $\zeta_{3} + 1$ |