Basic invariants
Dimension: | $2$ |
Group: | $S_3\times C_3$ |
Conductor: | \(1800\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5^{2} \) |
Artin stem field: | Galois closure of 6.0.25920000.2 |
Galois orbit size: | $2$ |
Smallest permutation container: | $S_3\times C_3$ |
Parity: | odd |
Determinant: | 1.72.6t1.a.a |
Projective image: | $S_3$ |
Projective stem field: | Galois closure of 3.1.16200.2 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - 2x^{5} - x^{4} - 2x^{3} + 38x^{2} - 68x + 43 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: \( x^{2} + 33x + 2 \)
Roots:
$r_{ 1 }$ | $=$ | \( 4 a + 35 + \left(10 a + 3\right)\cdot 37 + \left(22 a + 32\right)\cdot 37^{2} + \left(25 a + 23\right)\cdot 37^{3} + \left(16 a + 19\right)\cdot 37^{4} + \left(4 a + 33\right)\cdot 37^{5} +O(37^{6})\) |
$r_{ 2 }$ | $=$ | \( 33 a + 14 + \left(26 a + 3\right)\cdot 37 + 14 a\cdot 37^{2} + \left(11 a + 30\right)\cdot 37^{3} + \left(20 a + 23\right)\cdot 37^{4} + \left(32 a + 34\right)\cdot 37^{5} +O(37^{6})\) |
$r_{ 3 }$ | $=$ | \( 8 a + \left(3 a + 23\right)\cdot 37 + \left(24 a + 11\right)\cdot 37^{2} + \left(28 a + 21\right)\cdot 37^{3} + \left(6 a + 22\right)\cdot 37^{4} + \left(a + 5\right)\cdot 37^{5} +O(37^{6})\) |
$r_{ 4 }$ | $=$ | \( 29 a + 32 + \left(33 a + 27\right)\cdot 37 + \left(12 a + 30\right)\cdot 37^{2} + 8 a\cdot 37^{3} + \left(30 a + 21\right)\cdot 37^{4} + \left(35 a + 3\right)\cdot 37^{5} +O(37^{6})\) |
$r_{ 5 }$ | $=$ | \( 31 a + 28 + 29 a\cdot 37 + \left(28 a + 31\right)\cdot 37^{2} + \left(29 a + 27\right)\cdot 37^{3} + \left(24 a + 32\right)\cdot 37^{4} + \left(10 a + 7\right)\cdot 37^{5} +O(37^{6})\) |
$r_{ 6 }$ | $=$ | \( 6 a + 4 + \left(7 a + 15\right)\cdot 37 + \left(8 a + 5\right)\cdot 37^{2} + \left(7 a + 7\right)\cdot 37^{3} + \left(12 a + 28\right)\cdot 37^{4} + \left(26 a + 25\right)\cdot 37^{5} +O(37^{6})\) |
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 |
$1$ | $1$ | $()$ | $2$ |
$3$ | $2$ | $(1,4)(2,5)(3,6)$ | $0$ |
$1$ | $3$ | $(1,3,5)(2,4,6)$ | $2 \zeta_{3}$ |
$1$ | $3$ | $(1,5,3)(2,6,4)$ | $-2 \zeta_{3} - 2$ |
$2$ | $3$ | $(1,3,5)$ | $\zeta_{3} + 1$ |
$2$ | $3$ | $(1,5,3)$ | $-\zeta_{3}$ |
$2$ | $3$ | $(1,5,3)(2,4,6)$ | $-1$ |
$3$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
$3$ | $6$ | $(1,6,5,4,3,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.