Basic invariants
Dimension: | $2$ |
Group: | $S_3\times C_3$ |
Conductor: | \(2997\)\(\medspace = 3^{4} \cdot 37 \) |
Artin number field: | Galois closure of 6.0.26946027.2 |
Galois orbit size: | $2$ |
Smallest permutation container: | $S_3\times C_3$ |
Parity: | odd |
Projective image: | $S_3$ |
Projective field: | Galois closure of 3.1.4107.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$:
\( x^{2} + 58x + 2 \)
Roots:
$r_{ 1 }$ | $=$ | \( 36 a + 39 + \left(19 a + 35\right)\cdot 59 + \left(56 a + 28\right)\cdot 59^{2} + \left(23 a + 9\right)\cdot 59^{3} + \left(21 a + 17\right)\cdot 59^{4} + \left(16 a + 32\right)\cdot 59^{5} + 16 a\cdot 59^{6} +O(59^{7})\) |
$r_{ 2 }$ | $=$ | \( 51 a + 57 + \left(49 a + 2\right)\cdot 59 + \left(40 a + 20\right)\cdot 59^{2} + 35 a\cdot 59^{3} + \left(28 a + 56\right)\cdot 59^{4} + \left(2 a + 16\right)\cdot 59^{5} + \left(40 a + 7\right)\cdot 59^{6} +O(59^{7})\) |
$r_{ 3 }$ | $=$ | \( 15 a + 30 + \left(30 a + 21\right)\cdot 59 + \left(43 a + 19\right)\cdot 59^{2} + \left(11 a + 54\right)\cdot 59^{3} + \left(7 a + 51\right)\cdot 59^{4} + \left(45 a + 35\right)\cdot 59^{5} + \left(23 a + 13\right)\cdot 59^{6} +O(59^{7})\) |
$r_{ 4 }$ | $=$ | \( 44 a + 45 + \left(28 a + 36\right)\cdot 59 + \left(15 a + 32\right)\cdot 59^{2} + \left(47 a + 22\right)\cdot 59^{3} + \left(51 a + 47\right)\cdot 59^{4} + \left(13 a + 14\right)\cdot 59^{5} + \left(35 a + 51\right)\cdot 59^{6} +O(59^{7})\) |
$r_{ 5 }$ | $=$ | \( 8 a + 49 + \left(9 a + 1\right)\cdot 59 + \left(18 a + 11\right)\cdot 59^{2} + \left(23 a + 54\right)\cdot 59^{3} + \left(30 a + 48\right)\cdot 59^{4} + \left(56 a + 49\right)\cdot 59^{5} + \left(18 a + 44\right)\cdot 59^{6} +O(59^{7})\) |
$r_{ 6 }$ | $=$ | \( 23 a + 16 + \left(39 a + 19\right)\cdot 59 + \left(2 a + 6\right)\cdot 59^{2} + \left(35 a + 36\right)\cdot 59^{3} + \left(37 a + 14\right)\cdot 59^{4} + \left(42 a + 27\right)\cdot 59^{5} + 42 a\cdot 59^{6} +O(59^{7})\) |
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 | |
$c1$ | $c2$ | |||
$1$ | $1$ | $()$ | $2$ | $2$ |
$3$ | $2$ | $(1,2)(3,6)(4,5)$ | $0$ | $0$ |
$1$ | $3$ | $(1,5,3)(2,4,6)$ | $2 \zeta_{3}$ | $-2 \zeta_{3} - 2$ |
$1$ | $3$ | $(1,3,5)(2,6,4)$ | $-2 \zeta_{3} - 2$ | $2 \zeta_{3}$ |
$2$ | $3$ | $(1,3,5)$ | $-\zeta_{3}$ | $\zeta_{3} + 1$ |
$2$ | $3$ | $(1,5,3)$ | $\zeta_{3} + 1$ | $-\zeta_{3}$ |
$2$ | $3$ | $(1,3,5)(2,4,6)$ | $-1$ | $-1$ |
$3$ | $6$ | $(1,6,5,2,3,4)$ | $0$ | $0$ |
$3$ | $6$ | $(1,4,3,2,5,6)$ | $0$ | $0$ |