Basic invariants
| Dimension: | $5$ |
| Group: | $S_6$ |
| Conductor: | \(37253\) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.37253.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_6$ |
| Parity: | even |
| Determinant: | 1.37253.2t1.a.a |
| Projective image: | $S_6$ |
| Projective stem field: | Galois closure of 6.2.37253.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{4} + x^{2} - x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 113 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 113 }$:
\( x^{2} + 101x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 108 + 99\cdot 113 + 100\cdot 113^{2} + 76\cdot 113^{3} + 17\cdot 113^{4} +O(113^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 89 a + 68 + \left(104 a + 42\right)\cdot 113 + \left(44 a + 85\right)\cdot 113^{2} + \left(89 a + 12\right)\cdot 113^{3} + \left(94 a + 68\right)\cdot 113^{4} +O(113^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 24 a + 6 + \left(8 a + 81\right)\cdot 113 + \left(68 a + 67\right)\cdot 113^{2} + \left(23 a + 23\right)\cdot 113^{3} + \left(18 a + 99\right)\cdot 113^{4} +O(113^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 64 + 97\cdot 113 + 34\cdot 113^{2} + 46\cdot 113^{3} + 17\cdot 113^{4} +O(113^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 71 + 17\cdot 113 + 2\cdot 113^{2} + 44\cdot 113^{3} + 69\cdot 113^{4} +O(113^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 22 + 48\cdot 113^{2} + 22\cdot 113^{3} + 67\cdot 113^{4} +O(113^{5})\)
|
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.