Basic invariants
| Dimension: | $10$ |
| Group: | $S_6$ |
| Conductor: | \(8303765625000000\)\(\medspace = 2^{6} \cdot 3^{12} \cdot 5^{12} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.9112500.1 |
| 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 6.2.9112500.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 3x^{5} + 5x^{3} - 5 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 157 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 157 }$:
\( x^{2} + 152x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 119 + 129\cdot 157 + 27\cdot 157^{2} + 55\cdot 157^{3} + 65\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 137 + 77\cdot 157 + 64\cdot 157^{2} + 115\cdot 157^{3} + 25\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 139 + 29\cdot 157 + 32\cdot 157^{2} + 97\cdot 157^{3} + 6\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 156 a + 20 + \left(84 a + 66\right)\cdot 157 + \left(15 a + 146\right)\cdot 157^{2} + \left(56 a + 142\right)\cdot 157^{3} + \left(55 a + 78\right)\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( a + 15 + \left(72 a + 21\right)\cdot 157 + \left(141 a + 139\right)\cdot 157^{2} + \left(100 a + 93\right)\cdot 157^{3} + \left(101 a + 142\right)\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 44 + 146\cdot 157 + 60\cdot 157^{2} + 123\cdot 157^{3} + 151\cdot 157^{4} +O(157^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $10$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $2$ |
| $15$ | $2$ | $(1,2)$ | $-2$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $-2$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $1$ |
| $40$ | $3$ | $(1,2,3)$ | $1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $90$ | $4$ | $(1,2,3,4)$ | $0$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $-1$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.