Basic invariants
| Dimension: | $5$ |
| Group: | $S_6$ |
| Conductor: | \(37463\) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.37463.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_6$ |
| Parity: | odd |
| Determinant: | 1.37463.2t1.a.a |
| Projective image: | $S_6$ |
| Projective stem field: | Galois closure of 6.0.37463.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{4} - x^{3} + 2x^{2} + 2x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 167 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 167 }$:
\( x^{2} + 166x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 87 + 12\cdot 167 + 50\cdot 167^{2} + 43\cdot 167^{3} + 126\cdot 167^{4} +O(167^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 164 a + 153 + \left(76 a + 64\right)\cdot 167 + \left(157 a + 3\right)\cdot 167^{2} + \left(102 a + 165\right)\cdot 167^{3} + \left(12 a + 120\right)\cdot 167^{4} +O(167^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 57 a + 23 + \left(21 a + 63\right)\cdot 167 + \left(41 a + 142\right)\cdot 167^{2} + \left(47 a + 1\right)\cdot 167^{3} + \left(16 a + 42\right)\cdot 167^{4} +O(167^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 8 + 21\cdot 167 + 59\cdot 167^{2} + 5\cdot 167^{3} + 3\cdot 167^{4} +O(167^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 3 a + 150 + \left(90 a + 144\right)\cdot 167 + \left(9 a + 83\right)\cdot 167^{2} + \left(64 a + 110\right)\cdot 167^{3} + \left(154 a + 30\right)\cdot 167^{4} +O(167^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 110 a + 80 + \left(145 a + 27\right)\cdot 167 + \left(125 a + 162\right)\cdot 167^{2} + \left(119 a + 7\right)\cdot 167^{3} + \left(150 a + 11\right)\cdot 167^{4} +O(167^{5})\)
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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 |
| $1$ | $1$ | $()$ | $5$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-1$ |
| $15$ | $2$ | $(1,2)$ | $3$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $40$ | $3$ | $(1,2,3)$ | $2$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)$ | $1$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $-1$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.