Basic invariants
Dimension: | $10$ |
Group: | $S_6$ |
Conductor: | \(33364831591822329\)\(\medspace = 3^{14} \cdot 17^{8} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.0.20295603.1 |
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 6.0.20295603.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - x^{5} - 2x^{4} + x^{3} + 14x^{2} - 17x + 6 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 193 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 193 }$: \( x^{2} + 192x + 5 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 67 + 139\cdot 193 + 134\cdot 193^{2} + 101\cdot 193^{3} + 74\cdot 193^{4} +O(193^{5})\)
$r_{ 2 }$ |
$=$ |
\( 98 + 189\cdot 193 + 100\cdot 193^{2} + 118\cdot 193^{3} + 21\cdot 193^{4} +O(193^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 176 a + 155 + \left(141 a + 115\right)\cdot 193 + \left(138 a + 43\right)\cdot 193^{2} + \left(172 a + 146\right)\cdot 193^{3} + \left(105 a + 112\right)\cdot 193^{4} +O(193^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 98 a + 12 + \left(186 a + 175\right)\cdot 193 + \left(140 a + 55\right)\cdot 193^{2} + \left(98 a + 37\right)\cdot 193^{3} + \left(59 a + 85\right)\cdot 193^{4} +O(193^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 95 a + 110 + \left(6 a + 70\right)\cdot 193 + \left(52 a + 10\right)\cdot 193^{2} + \left(94 a + 188\right)\cdot 193^{3} + \left(133 a + 45\right)\cdot 193^{4} +O(193^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 17 a + 138 + \left(51 a + 81\right)\cdot 193 + \left(54 a + 40\right)\cdot 193^{2} + \left(20 a + 180\right)\cdot 193^{3} + \left(87 a + 45\right)\cdot 193^{4} +O(193^{5})\)
| |
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.