Properties

Label 5.50173.6t16.a.a
Dimension $5$
Group $S_6$
Conductor $50173$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $5$
Group: $S_6$
Conductor: \(50173\)\(\medspace = 131 \cdot 383 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.2.50173.1
Galois orbit size: $1$
Smallest permutation container: $S_6$
Parity: even
Determinant: 1.50173.2t1.a.a
Projective image: $S_6$
Projective stem field: Galois closure of 6.2.50173.1

Defining polynomial

$f(x)$$=$ \( x^{6} - x^{5} + x^{4} + 2x^{3} - 4x^{2} + 3x - 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 97 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 97 }$: \( x^{2} + 96x + 5 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 5 + 91\cdot 97 + 75\cdot 97^{2} + 36\cdot 97^{4} +O(97^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 66 a + 3 + \left(28 a + 43\right)\cdot 97 + \left(16 a + 75\right)\cdot 97^{2} + \left(70 a + 66\right)\cdot 97^{3} + \left(26 a + 34\right)\cdot 97^{4} +O(97^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 31 + 12\cdot 97 + 72\cdot 97^{2} + 47\cdot 97^{3} + 92\cdot 97^{4} +O(97^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 87 a + 31 a\cdot 97 + \left(a + 66\right)\cdot 97^{2} + \left(54 a + 49\right)\cdot 97^{3} + \left(71 a + 59\right)\cdot 97^{4} +O(97^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 10 a + 87 + \left(65 a + 41\right)\cdot 97 + \left(95 a + 35\right)\cdot 97^{2} + \left(42 a + 5\right)\cdot 97^{3} + \left(25 a + 77\right)\cdot 97^{4} +O(97^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 31 a + 69 + \left(68 a + 5\right)\cdot 97 + \left(80 a + 63\right)\cdot 97^{2} + \left(26 a + 23\right)\cdot 97^{3} + \left(70 a + 88\right)\cdot 97^{4} +O(97^{5})\) 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.