Basic invariants
| Dimension: | $16$ |
| Group: | $S_6$ |
| Conductor: | \(843\!\cdots\!176\)\(\medspace = 2^{12} \cdot 3461^{8} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.55376.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 36T1252 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_6$ |
| Projective stem field: | Galois closure of 6.2.55376.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - 2x^{4} + 3x^{3} - x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 149 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 149 }$:
\( x^{2} + 145x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 64 + 134\cdot 149 + 21\cdot 149^{2} + 46\cdot 149^{3} + 69\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 65 + 122\cdot 149 + 70\cdot 149^{2} + 106\cdot 149^{3} + 23\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 95 a + 31 + \left(51 a + 53\right)\cdot 149 + \left(27 a + 47\right)\cdot 149^{2} + \left(102 a + 24\right)\cdot 149^{3} + \left(115 a + 42\right)\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 15 + 47\cdot 149 + 45\cdot 149^{2} + 43\cdot 149^{3} + 107\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 54 a + 113 + \left(97 a + 15\right)\cdot 149 + \left(121 a + 105\right)\cdot 149^{2} + \left(46 a + 107\right)\cdot 149^{3} + \left(33 a + 104\right)\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 11 + 74\cdot 149 + 7\cdot 149^{2} + 119\cdot 149^{3} + 99\cdot 149^{4} +O(149^{5})\)
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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 |
| $1$ | $1$ | $()$ | $16$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $15$ | $2$ | $(1,2)$ | $0$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-2$ |
| $40$ | $3$ | $(1,2,3)$ | $-2$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $90$ | $4$ | $(1,2,3,4)$ | $0$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.