Basic invariants
Dimension: | $2$ |
Group: | $D_{6}$ |
Conductor: | \(3724\)\(\medspace = 2^{2} \cdot 7^{2} \cdot 19 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 6.0.1981168.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{6}$ |
Parity: | odd |
Projective image: | $S_3$ |
Projective field: | Galois closure of 3.1.76.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{2} + 24x + 2 \)
Roots:
$r_{ 1 }$ | $=$ | \( a + \left(19 a + 1\right)\cdot 29 + 26 a\cdot 29^{2} + \left(23 a + 8\right)\cdot 29^{3} + \left(25 a + 5\right)\cdot 29^{4} + \left(20 a + 26\right)\cdot 29^{5} +O(29^{6})\) |
$r_{ 2 }$ | $=$ | \( 28 a + 5 + \left(9 a + 8\right)\cdot 29 + \left(2 a + 27\right)\cdot 29^{2} + \left(5 a + 13\right)\cdot 29^{3} + \left(3 a + 23\right)\cdot 29^{4} + \left(8 a + 17\right)\cdot 29^{5} +O(29^{6})\) |
$r_{ 3 }$ | $=$ | \( a + 14 + \left(19 a + 8\right)\cdot 29 + \left(26 a + 27\right)\cdot 29^{2} + \left(23 a + 8\right)\cdot 29^{3} + \left(25 a + 23\right)\cdot 29^{4} + \left(20 a + 2\right)\cdot 29^{5} +O(29^{6})\) |
$r_{ 4 }$ | $=$ | \( 3 + 16\cdot 29 + 2\cdot 29^{2} + 21\cdot 29^{3} + 5\cdot 29^{4} + 11\cdot 29^{5} +O(29^{6})\) |
$r_{ 5 }$ | $=$ | \( 28 a + 19 + \left(9 a + 15\right)\cdot 29 + \left(2 a + 25\right)\cdot 29^{2} + \left(5 a + 14\right)\cdot 29^{3} + \left(3 a + 12\right)\cdot 29^{4} + \left(8 a + 23\right)\cdot 29^{5} +O(29^{6})\) |
$r_{ 6 }$ | $=$ | \( 18 + 8\cdot 29 + 4\cdot 29^{2} + 20\cdot 29^{3} + 16\cdot 29^{4} + 5\cdot 29^{5} +O(29^{6})\) |
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 values |
$c1$ | |||
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,3)(2,5)(4,6)$ | $-2$ |
$3$ | $2$ | $(2,6)(4,5)$ | $0$ |
$3$ | $2$ | $(1,3)(2,4)(5,6)$ | $0$ |
$2$ | $3$ | $(1,2,6)(3,5,4)$ | $-1$ |
$2$ | $6$ | $(1,5,6,3,2,4)$ | $1$ |