Basic invariants
| Dimension: | $3$ |
| Group: | $S_4\times C_2$ |
| Conductor: | \(4260\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 71 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.6049200.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4\times C_2$ |
| Parity: | even |
| Determinant: | 1.1065.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.12780.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 3x^{5} + 8x^{4} - 11x^{3} + 8x^{2} - 3x + 11 \)
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The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$:
\( x^{2} + 33x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 8 + 24\cdot 37 + 35\cdot 37^{2} + 2\cdot 37^{3} + 36\cdot 37^{4} + 9\cdot 37^{5} + 8\cdot 37^{6} + 18\cdot 37^{7} + 21\cdot 37^{8} +O(37^{9})\)
|
| $r_{ 2 }$ | $=$ |
\( 15 a + \left(35 a + 20\right)\cdot 37 + \left(22 a + 30\right)\cdot 37^{2} + \left(21 a + 19\right)\cdot 37^{3} + \left(12 a + 17\right)\cdot 37^{4} + \left(17 a + 4\right)\cdot 37^{5} + \left(14 a + 29\right)\cdot 37^{6} + \left(13 a + 18\right)\cdot 37^{7} + \left(2 a + 3\right)\cdot 37^{8} +O(37^{9})\)
|
| $r_{ 3 }$ | $=$ |
\( 22 a + 23 + \left(a + 35\right)\cdot 37 + \left(14 a + 12\right)\cdot 37^{2} + \left(15 a + 9\right)\cdot 37^{3} + \left(24 a + 9\right)\cdot 37^{4} + \left(19 a + 24\right)\cdot 37^{5} + \left(22 a + 32\right)\cdot 37^{6} + \left(23 a + 20\right)\cdot 37^{7} + \left(34 a + 36\right)\cdot 37^{8} +O(37^{9})\)
|
| $r_{ 4 }$ | $=$ |
\( 30 + 12\cdot 37 + 37^{2} + 34\cdot 37^{3} + 27\cdot 37^{5} + 28\cdot 37^{6} + 18\cdot 37^{7} + 15\cdot 37^{8} +O(37^{9})\)
|
| $r_{ 5 }$ | $=$ |
\( 15 a + 15 + \left(35 a + 1\right)\cdot 37 + \left(22 a + 24\right)\cdot 37^{2} + \left(21 a + 27\right)\cdot 37^{3} + \left(12 a + 27\right)\cdot 37^{4} + \left(17 a + 12\right)\cdot 37^{5} + \left(14 a + 4\right)\cdot 37^{6} + \left(13 a + 16\right)\cdot 37^{7} + 2 a\cdot 37^{8} +O(37^{9})\)
|
| $r_{ 6 }$ | $=$ |
\( 22 a + 1 + \left(a + 17\right)\cdot 37 + \left(14 a + 6\right)\cdot 37^{2} + \left(15 a + 17\right)\cdot 37^{3} + \left(24 a + 19\right)\cdot 37^{4} + \left(19 a + 32\right)\cdot 37^{5} + \left(22 a + 7\right)\cdot 37^{6} + \left(23 a + 18\right)\cdot 37^{7} + \left(34 a + 33\right)\cdot 37^{8} +O(37^{9})\)
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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 | Complex conjugation |
| $1$ | $1$ | $()$ | $3$ | |
| $1$ | $2$ | $(1,4)(2,6)(3,5)$ | $-3$ | |
| $3$ | $2$ | $(3,5)$ | $1$ | |
| $3$ | $2$ | $(1,4)(3,5)$ | $-1$ | |
| $6$ | $2$ | $(1,2)(4,6)$ | $1$ | |
| $6$ | $2$ | $(1,2)(3,5)(4,6)$ | $-1$ | ✓ |
| $8$ | $3$ | $(1,2,3)(4,6,5)$ | $0$ | |
| $6$ | $4$ | $(1,3,4,5)$ | $1$ | |
| $6$ | $4$ | $(1,4)(2,3,6,5)$ | $-1$ | |
| $8$ | $6$ | $(1,2,3,4,6,5)$ | $0$ |