Basic invariants
| Dimension: | $10$ |
| Group: | $S_6$ |
| Conductor: | \(7834102237495296\)\(\medspace = 2^{24} \cdot 3^{4} \cdot 7^{8} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.3687936.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 30T164 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_6$ |
| Projective stem field: | Galois closure of 6.2.3687936.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 4x^{4} - 2x^{3} + x^{2} - 2x - 5 \)
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The roots of $f$ are computed in an extension of $\Q_{ 193 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 193 }$:
\( x^{2} + 192x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 167 a + 52 + \left(186 a + 78\right)\cdot 193 + \left(190 a + 25\right)\cdot 193^{2} + \left(157 a + 90\right)\cdot 193^{3} + \left(112 a + 99\right)\cdot 193^{4} +O(193^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 49 + 23\cdot 193 + 35\cdot 193^{2} + 151\cdot 193^{3} + 93\cdot 193^{4} +O(193^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 26 a + 26 + \left(6 a + 98\right)\cdot 193 + \left(2 a + 29\right)\cdot 193^{2} + \left(35 a + 57\right)\cdot 193^{3} + \left(80 a + 54\right)\cdot 193^{4} +O(193^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 62 + 104\cdot 193 + 147\cdot 193^{2} + 13\cdot 193^{3} + 32\cdot 193^{4} +O(193^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 134 a + 128 + \left(57 a + 175\right)\cdot 193 + \left(141 a + 128\right)\cdot 193^{2} + \left(69 a + 72\right)\cdot 193^{3} + \left(149 a + 13\right)\cdot 193^{4} +O(193^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 59 a + 69 + \left(135 a + 99\right)\cdot 193 + \left(51 a + 19\right)\cdot 193^{2} + \left(123 a + 1\right)\cdot 193^{3} + \left(43 a + 93\right)\cdot 193^{4} +O(193^{5})\)
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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 |
| $1$ | $1$ | $()$ | $10$ | |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-2$ | |
| $15$ | $2$ | $(1,2)$ | $2$ | |
| $45$ | $2$ | $(1,2)(3,4)$ | $-2$ | ✓ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $1$ | |
| $40$ | $3$ | $(1,2,3)$ | $1$ | |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ | |
| $90$ | $4$ | $(1,2,3,4)$ | $0$ | |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ | |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $1$ | |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |