Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(3300\)\(\medspace = 2^{2} \cdot 3 \cdot 5^{2} \cdot 11 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 6.0.217800000.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Projective image: | $S_3$ |
| Projective field: | Galois closure of 3.1.3300.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$:
\( x^{2} + 60x + 2 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 58 a + 23 + \left(2 a + 28\right)\cdot 61 + \left(46 a + 46\right)\cdot 61^{2} + \left(32 a + 40\right)\cdot 61^{3} + \left(44 a + 58\right)\cdot 61^{4} + \left(36 a + 30\right)\cdot 61^{5} +O(61^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 24 + 14\cdot 61 + 7\cdot 61^{2} + 4\cdot 61^{3} + 3\cdot 61^{4} + 46\cdot 61^{5} +O(61^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 58 a + 41 + \left(2 a + 26\right)\cdot 61 + \left(46 a + 32\right)\cdot 61^{2} + \left(32 a + 33\right)\cdot 61^{3} + \left(44 a + 51\right)\cdot 61^{4} + \left(36 a + 37\right)\cdot 61^{5} +O(61^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 3 a + 38 + \left(58 a + 32\right)\cdot 61 + \left(14 a + 14\right)\cdot 61^{2} + \left(28 a + 20\right)\cdot 61^{3} + \left(16 a + 2\right)\cdot 61^{4} + \left(24 a + 30\right)\cdot 61^{5} +O(61^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 37 + 46\cdot 61 + 53\cdot 61^{2} + 56\cdot 61^{3} + 57\cdot 61^{4} + 14\cdot 61^{5} +O(61^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 3 a + 20 + \left(58 a + 34\right)\cdot 61 + \left(14 a + 28\right)\cdot 61^{2} + \left(28 a + 27\right)\cdot 61^{3} + \left(16 a + 9\right)\cdot 61^{4} + \left(24 a + 23\right)\cdot 61^{5} +O(61^{6})\)
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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 values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-2$ |
| $3$ | $2$ | $(2,3)(5,6)$ | $0$ |
| $3$ | $2$ | $(1,2)(3,6)(4,5)$ | $0$ |
| $2$ | $3$ | $(1,6,5)(2,4,3)$ | $-1$ |
| $2$ | $6$ | $(1,2,6,4,5,3)$ | $1$ |