Properties

Label 6.543...469.20t30.a.a
Dimension $6$
Group $S_5$
Conductor $5.431\times 10^{14}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $6$
Group: $S_5$
Conductor: \(543118793139469\)\(\medspace = 83^{3} \cdot 983^{3} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 5.5.81589.1
Galois orbit size: $1$
Smallest permutation container: 20T30
Parity: even
Determinant: 1.81589.2t1.a.a
Projective image: $S_5$
Projective stem field: Galois closure of 5.5.81589.1

Defining polynomial

$f(x)$$=$ \( x^{5} - 6x^{3} + 8x - 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: \( x^{2} + 12x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 12 a + 6 + 5\cdot 13 + 4 a\cdot 13^{2} + \left(5 a + 11\right)\cdot 13^{3} + \left(3 a + 7\right)\cdot 13^{4} +O(13^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 7 a + 9 + \left(4 a + 8\right)\cdot 13 + \left(12 a + 9\right)\cdot 13^{2} + \left(2 a + 4\right)\cdot 13^{3} + 8\cdot 13^{4} +O(13^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 3 + 11\cdot 13 + 7\cdot 13^{2} + 2\cdot 13^{3} + 11\cdot 13^{4} +O(13^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( a + 5 + \left(12 a + 7\right)\cdot 13 + \left(8 a + 3\right)\cdot 13^{2} + \left(7 a + 12\right)\cdot 13^{3} + \left(9 a + 5\right)\cdot 13^{4} +O(13^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 6 a + 3 + \left(8 a + 6\right)\cdot 13 + 4\cdot 13^{2} + \left(10 a + 8\right)\cdot 13^{3} + \left(12 a + 5\right)\cdot 13^{4} +O(13^{5})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 5 }$

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 5 }$ Character value
$1$$1$$()$$6$
$10$$2$$(1,2)$$0$
$15$$2$$(1,2)(3,4)$$-2$
$20$$3$$(1,2,3)$$0$
$30$$4$$(1,2,3,4)$$0$
$24$$5$$(1,2,3,4,5)$$1$
$20$$6$$(1,2,3)(4,5)$$0$

The blue line marks the conjugacy class containing complex conjugation.