Properties

Label 5.37423.6t16.a.a
Dimension $5$
Group $S_6$
Conductor $37423$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $5$
Group: $S_6$
Conductor: \(37423\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.0.37423.1
Galois orbit size: $1$
Smallest permutation container: $S_6$
Parity: odd
Determinant: 1.37423.2t1.a.a
Projective image: $S_6$
Projective stem field: Galois closure of 6.0.37423.1

Defining polynomial

$f(x)$$=$ \( x^{6} - x^{5} + x^{4} + 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: \( x^{2} + 38x + 6 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 22 a + 23 + \left(17 a + 39\right)\cdot 41 + \left(5 a + 6\right)\cdot 41^{2} + \left(36 a + 10\right)\cdot 41^{3} + \left(17 a + 37\right)\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 6 a + 6 + \left(13 a + 10\right)\cdot 41 + \left(12 a + 4\right)\cdot 41^{2} + \left(37 a + 21\right)\cdot 41^{3} + \left(5 a + 9\right)\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 19 a + 7 + \left(23 a + 29\right)\cdot 41 + \left(35 a + 5\right)\cdot 41^{2} + \left(4 a + 31\right)\cdot 41^{3} + \left(23 a + 13\right)\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 34 + 20\cdot 41 + 2\cdot 41^{2} + 30\cdot 41^{3} + 10\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 30 + 20\cdot 41 + 34\cdot 41^{2} + 32\cdot 41^{3} + 20\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 35 a + 24 + \left(27 a + 2\right)\cdot 41 + \left(28 a + 28\right)\cdot 41^{2} + \left(3 a + 38\right)\cdot 41^{3} + \left(35 a + 30\right)\cdot 41^{4} +O(41^{5})\) 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 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.