Properties

Label 5.52891.6t16.a
Dimension $5$
Group $S_6$
Conductor $52891$
Indicator $1$

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Basic invariants

Dimension:$5$
Group:$S_6$
Conductor:\(52891\)\(\medspace = 227 \cdot 233 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 6.0.52891.1
Galois orbit size: $1$
Smallest permutation container: $S_6$
Parity: odd
Projective image: $S_6$
Projective field: Galois closure of 6.0.52891.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 131 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 131 }$: \( x^{2} + 127x + 2 \) Copy content Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 16 a + 1 + \left(106 a + 42\right)\cdot 131 + \left(66 a + 50\right)\cdot 131^{2} + \left(104 a + 109\right)\cdot 131^{3} + \left(76 a + 127\right)\cdot 131^{4} +O(131^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 108 + 112\cdot 131 + 5\cdot 131^{2} + 120\cdot 131^{3} + 68\cdot 131^{4} +O(131^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 38 a + 87 + \left(76 a + 125\right)\cdot 131 + \left(81 a + 101\right)\cdot 131^{2} + \left(76 a + 12\right)\cdot 131^{3} + \left(119 a + 5\right)\cdot 131^{4} +O(131^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 25 + 55\cdot 131 + 64\cdot 131^{2} + 107\cdot 131^{3} + 108\cdot 131^{4} +O(131^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 93 a + 108 + \left(54 a + 130\right)\cdot 131 + \left(49 a + 89\right)\cdot 131^{2} + \left(54 a + 106\right)\cdot 131^{3} + \left(11 a + 13\right)\cdot 131^{4} +O(131^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 115 a + 65 + \left(24 a + 57\right)\cdot 131 + \left(64 a + 80\right)\cdot 131^{2} + \left(26 a + 67\right)\cdot 131^{3} + \left(54 a + 68\right)\cdot 131^{4} +O(131^{5})\) Copy content Toggle raw display

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

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character values
$c1$
$1$ $1$ $()$ $5$
$15$ $2$ $(1,2)(3,4)(5,6)$ $-1$
$15$ $2$ $(1,2)$ $3$
$45$ $2$ $(1,2)(3,4)$ $1$
$40$ $3$ $(1,2,3)(4,5,6)$ $-1$
$40$ $3$ $(1,2,3)$ $2$
$90$ $4$ $(1,2,3,4)(5,6)$ $-1$
$90$ $4$ $(1,2,3,4)$ $1$
$144$ $5$ $(1,2,3,4,5)$ $0$
$120$ $6$ $(1,2,3,4,5,6)$ $-1$
$120$ $6$ $(1,2,3)(4,5)$ $0$
The blue line marks the conjugacy class containing complex conjugation.