Properties

Label 5.127168.6t16.a
Dimension $5$
Group $S_6$
Conductor $127168$
Indicator $1$

Related objects

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Basic invariants

Dimension:$5$
Group:$S_6$
Conductor:\(127168\)\(\medspace = 2^{6} \cdot 1987 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 6.2.127168.1
Galois orbit size: $1$
Smallest permutation container: $S_6$
Parity: even
Projective image: $S_6$
Projective field: 6.2.127168.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: \(x^{2} + 24 x + 2\)  Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 13 a + 28 + \left(25 a + 4\right)\cdot 29 + \left(18 a + 20\right)\cdot 29^{2} + \left(23 a + 19\right)\cdot 29^{3} + 24\cdot 29^{4} +O(29^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 16 a + 6 + \left(3 a + 3\right)\cdot 29 + \left(10 a + 2\right)\cdot 29^{2} + \left(5 a + 3\right)\cdot 29^{3} + \left(28 a + 5\right)\cdot 29^{4} +O(29^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 10 a + 26 + \left(4 a + 9\right)\cdot 29 + \left(8 a + 19\right)\cdot 29^{2} + \left(12 a + 19\right)\cdot 29^{3} + 4\cdot 29^{4} +O(29^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 19 a + 18 + \left(24 a + 21\right)\cdot 29 + \left(20 a + 26\right)\cdot 29^{2} + \left(16 a + 14\right)\cdot 29^{3} + \left(28 a + 23\right)\cdot 29^{4} +O(29^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 18 a + 4 + \left(16 a + 20\right)\cdot 29 + \left(21 a + 21\right)\cdot 29^{2} + \left(18 a + 7\right)\cdot 29^{3} + \left(8 a + 2\right)\cdot 29^{4} +O(29^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 11 a + 7 + \left(12 a + 27\right)\cdot 29 + \left(7 a + 25\right)\cdot 29^{2} + \left(10 a + 21\right)\cdot 29^{3} + \left(20 a + 26\right)\cdot 29^{4} +O(29^{5})\)  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.