Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(8303\)\(\medspace = 19^{2} \cdot 23 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.83453453.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Determinant: | 1.23.2t1.a.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.23.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + 11x^{4} - 53x^{3} + 68x^{2} - 215x - 2269 \)
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The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$:
\( x^{2} + 33x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 32 a + 6 + \left(33 a + 4\right)\cdot 37 + \left(6 a + 29\right)\cdot 37^{2} + \left(11 a + 7\right)\cdot 37^{3} + \left(36 a + 31\right)\cdot 37^{4} + \left(35 a + 18\right)\cdot 37^{5} + \left(15 a + 30\right)\cdot 37^{6} +O(37^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 5 a + 23 + \left(3 a + 33\right)\cdot 37 + \left(30 a + 22\right)\cdot 37^{2} + \left(25 a + 8\right)\cdot 37^{3} + 17\cdot 37^{4} + \left(a + 15\right)\cdot 37^{5} + \left(21 a + 21\right)\cdot 37^{6} +O(37^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 9 + 36\cdot 37 + 21\cdot 37^{2} + 20\cdot 37^{3} + 25\cdot 37^{4} + 2\cdot 37^{5} + 22\cdot 37^{6} +O(37^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 33 a + 21 + \left(3 a + 31\right)\cdot 37 + \left(34 a + 17\right)\cdot 37^{2} + \left(4 a + 19\right)\cdot 37^{3} + \left(28 a + 4\right)\cdot 37^{4} + \left(10 a + 35\right)\cdot 37^{5} + 32 a\cdot 37^{6} +O(37^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 12 + 28\cdot 37 + 16\cdot 37^{2} + 12\cdot 37^{3} + 31\cdot 37^{4} + 25\cdot 37^{5} + 27\cdot 37^{6} +O(37^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 4 a + 5 + \left(33 a + 14\right)\cdot 37 + \left(2 a + 2\right)\cdot 37^{2} + \left(32 a + 5\right)\cdot 37^{3} + \left(8 a + 1\right)\cdot 37^{4} + \left(26 a + 13\right)\cdot 37^{5} + \left(4 a + 8\right)\cdot 37^{6} +O(37^{7})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,6)(3,5)$ | $-2$ |
| $3$ | $2$ | $(2,3)(5,6)$ | $0$ |
| $3$ | $2$ | $(1,4)(2,5)(3,6)$ | $0$ |
| $2$ | $3$ | $(1,2,3)(4,6,5)$ | $-1$ |
| $2$ | $6$ | $(1,6,3,4,2,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.