Basic invariants
| Dimension: | $3$ |
| Group: | $S_4\times C_2$ |
| Conductor: | \(80375\)\(\medspace = 5^{3} \cdot 643 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.33230963375.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4\times C_2$ |
| Parity: | odd |
| Determinant: | 1.3215.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.643.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 3x^{5} + 44x^{4} - 38x^{3} + 650x^{2} - 139x + 5489 \)
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The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$:
\( x^{2} + 33x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 12 a + 30 + \left(33 a + 10\right)\cdot 37 + \left(10 a + 25\right)\cdot 37^{2} + \left(30 a + 24\right)\cdot 37^{3} + \left(8 a + 16\right)\cdot 37^{4} + \left(26 a + 13\right)\cdot 37^{5} + \left(17 a + 34\right)\cdot 37^{6} + \left(34 a + 1\right)\cdot 37^{7} + \left(36 a + 22\right)\cdot 37^{8} + \left(11 a + 32\right)\cdot 37^{9} +O(37^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 19 a + 10 + \left(18 a + 2\right)\cdot 37 + \left(22 a + 17\right)\cdot 37^{2} + \left(31 a + 11\right)\cdot 37^{3} + \left(20 a + 24\right)\cdot 37^{4} + \left(22 a + 22\right)\cdot 37^{5} + \left(34 a + 28\right)\cdot 37^{6} + \left(4 a + 18\right)\cdot 37^{7} + 30\cdot 37^{8} + \left(8 a + 7\right)\cdot 37^{9} +O(37^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 36 + 18\cdot 37 + 23\cdot 37^{2} + 7\cdot 37^{4} + 25\cdot 37^{5} + 7\cdot 37^{6} + 2\cdot 37^{7} + 3\cdot 37^{8} + 33\cdot 37^{9} +O(37^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 22 + 32\cdot 37^{2} + 8\cdot 37^{3} + 2\cdot 37^{4} + 33\cdot 37^{5} + 37^{6} + 36\cdot 37^{7} + 4\cdot 37^{8} + 28\cdot 37^{9} +O(37^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 18 a + 12 + \left(18 a + 20\right)\cdot 37 + \left(14 a + 14\right)\cdot 37^{2} + \left(5 a + 4\right)\cdot 37^{3} + \left(16 a + 2\right)\cdot 37^{4} + \left(14 a + 18\right)\cdot 37^{5} + \left(2 a + 33\right)\cdot 37^{6} + \left(32 a + 3\right)\cdot 37^{7} + \left(36 a + 26\right)\cdot 37^{8} + \left(28 a + 2\right)\cdot 37^{9} +O(37^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 25 a + 4 + \left(3 a + 21\right)\cdot 37 + \left(26 a + 35\right)\cdot 37^{2} + \left(6 a + 23\right)\cdot 37^{3} + \left(28 a + 21\right)\cdot 37^{4} + \left(10 a + 35\right)\cdot 37^{5} + \left(19 a + 4\right)\cdot 37^{6} + \left(2 a + 11\right)\cdot 37^{7} + 24\cdot 37^{8} + \left(25 a + 6\right)\cdot 37^{9} +O(37^{10})\)
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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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,2)(3,4)(5,6)$ | $-3$ |
| $3$ | $2$ | $(5,6)$ | $1$ |
| $3$ | $2$ | $(3,4)(5,6)$ | $-1$ |
| $6$ | $2$ | $(1,3)(2,4)$ | $-1$ |
| $6$ | $2$ | $(1,3)(2,4)(5,6)$ | $1$ |
| $8$ | $3$ | $(1,5,3)(2,6,4)$ | $0$ |
| $6$ | $4$ | $(3,5,4,6)$ | $-1$ |
| $6$ | $4$ | $(1,2)(3,5,4,6)$ | $1$ |
| $8$ | $6$ | $(1,5,4,2,6,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.