Properties

Label 5.43531.6t16.a
Dimension $5$
Group $S_6$
Conductor $43531$
Indicator $1$

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Basic invariants

Dimension:$5$
Group:$S_6$
Conductor:\(43531\)\(\medspace = 101 \cdot 431 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 6.0.43531.1
Galois orbit size: $1$
Smallest permutation container: $S_6$
Parity: odd
Projective image: $S_6$
Projective field: Galois closure of 6.0.43531.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 157 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 157 }$: \( x^{2} + 152x + 5 \) Copy content Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 119 + 95\cdot 157 + 44\cdot 157^{2} + 72\cdot 157^{3} + 99\cdot 157^{4} +O(157^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 148 a + 40 + \left(95 a + 38\right)\cdot 157 + \left(94 a + 121\right)\cdot 157^{2} + \left(106 a + 72\right)\cdot 157^{3} + \left(131 a + 97\right)\cdot 157^{4} +O(157^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 59 + 63\cdot 157 + 5\cdot 157^{2} + 131\cdot 157^{3} + 46\cdot 157^{4} +O(157^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 25 a + 145 + \left(88 a + 57\right)\cdot 157 + \left(107 a + 68\right)\cdot 157^{2} + \left(82 a + 81\right)\cdot 157^{3} + \left(114 a + 93\right)\cdot 157^{4} +O(157^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 9 a + 152 + \left(61 a + 55\right)\cdot 157 + \left(62 a + 27\right)\cdot 157^{2} + \left(50 a + 40\right)\cdot 157^{3} + \left(25 a + 21\right)\cdot 157^{4} +O(157^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 132 a + 113 + \left(68 a + 2\right)\cdot 157 + \left(49 a + 47\right)\cdot 157^{2} + \left(74 a + 73\right)\cdot 157^{3} + \left(42 a + 112\right)\cdot 157^{4} +O(157^{5})\) 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.