Basic invariants
| Dimension: | $10$ |
| Group: | $S_6$ |
| Conductor: | \(30344516420400625\)\(\medspace = 5^{4} \cdot 191^{6} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.182405.1 |
| 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 6.2.182405.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{4} - 3x^{3} + x + 3 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$:
\( x^{2} + 70x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 40 + 15\cdot 73 + 43\cdot 73^{2} + 33\cdot 73^{3} + 30\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 47 + 27\cdot 73 + 48\cdot 73^{2} + 54\cdot 73^{3} + 59\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 61 a + 56 + \left(69 a + 68\right)\cdot 73 + \left(13 a + 32\right)\cdot 73^{2} + \left(32 a + 68\right)\cdot 73^{3} + \left(58 a + 5\right)\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 58 + 32\cdot 73 + 12\cdot 73^{3} + 31\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 71 + 2\cdot 73 + 16\cdot 73^{2} + 45\cdot 73^{3} + 15\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 12 a + 20 + \left(3 a + 71\right)\cdot 73 + \left(59 a + 4\right)\cdot 73^{2} + \left(40 a + 5\right)\cdot 73^{3} + \left(14 a + 3\right)\cdot 73^{4} +O(73^{5})\)
|
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.