Properties

 Label 5.8294400.12t183.b.a Dimension $5$ Group $S_6$ Conductor $8294400$ Root number $1$ Indicator $1$

Related objects

Basic invariants

 Dimension: $5$ Group: $S_6$ Conductor: $$8294400$$$$\medspace = 2^{12} \cdot 3^{4} \cdot 5^{2}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 6.2.1036800.1 Galois orbit size: $1$ Smallest permutation container: 12T183 Parity: even Determinant: 1.1.1t1.a.a Projective image: $S_6$ Projective field: Galois closure of 6.2.1036800.1

Defining polynomial

 $f(x)$ $=$ $x^{6} - 2 x^{5} - x^{4} + 6 x^{3} - 2 x^{2} - 4 x - 1$.

The roots of $f$ are computed in an extension of $\Q_{ 137 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 137 }$: $x^{2} + 131 x + 3$

Roots:
 $r_{ 1 }$ $=$ $118 a + 5 + \left(93 a + 77\right)\cdot 137 + \left(97 a + 36\right)\cdot 137^{2} + \left(50 a + 102\right)\cdot 137^{3} + \left(121 a + 19\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$ $r_{ 2 }$ $=$ $19 a + 28 + \left(43 a + 111\right)\cdot 137 + \left(39 a + 117\right)\cdot 137^{2} + \left(86 a + 34\right)\cdot 137^{3} + \left(15 a + 12\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$ $r_{ 3 }$ $=$ $51 + 99\cdot 137 + 51\cdot 137^{2} + 46\cdot 137^{3} + 97\cdot 137^{4} +O\left(137^{ 5 }\right)$ $r_{ 4 }$ $=$ $29 a + 88 + \left(21 a + 131\right)\cdot 137 + \left(51 a + 24\right)\cdot 137^{2} + \left(96 a + 68\right)\cdot 137^{3} + \left(91 a + 92\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$ $r_{ 5 }$ $=$ $116 + 35\cdot 137 + 6\cdot 137^{2} + 112\cdot 137^{3} + 53\cdot 137^{4} +O\left(137^{ 5 }\right)$ $r_{ 6 }$ $=$ $108 a + 125 + \left(115 a + 92\right)\cdot 137 + \left(85 a + 36\right)\cdot 137^{2} + \left(40 a + 47\right)\cdot 137^{3} + \left(45 a + 135\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

 Cycle notation $(1,2)$ $(1,2,3,4,5,6)$

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.