Properties

Label 5.969263689.12t183.a.a
Dimension $5$
Group $S_6$
Conductor $969263689$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $5$
Group: $S_6$
Conductor: \(969263689\)\(\medspace = 163^{2} \cdot 191^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.2.31133.1
Galois orbit size: $1$
Smallest permutation container: 12T183
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $S_6$
Projective stem field: Galois closure of 6.2.31133.1

Defining polynomial

$f(x)$$=$ \( x^{6} - 2x^{4} - x^{3} + 2x + 1 \) 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:
$r_{ 1 }$ $=$ \( 10 + 46\cdot 97 + 94\cdot 97^{2} + 64\cdot 97^{3} + 28\cdot 97^{4} +O(97^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 10 a + 33 + \left(28 a + 6\right)\cdot 97 + \left(12 a + 23\right)\cdot 97^{2} + \left(83 a + 45\right)\cdot 97^{3} + \left(11 a + 77\right)\cdot 97^{4} +O(97^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 21 a + 27 + \left(5 a + 2\right)\cdot 97 + \left(77 a + 2\right)\cdot 97^{2} + \left(81 a + 72\right)\cdot 97^{3} + \left(32 a + 35\right)\cdot 97^{4} +O(97^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 33 + 31\cdot 97 + 90\cdot 97^{2} + 12\cdot 97^{3} + 59\cdot 97^{4} +O(97^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 76 a + 48 + \left(91 a + 83\right)\cdot 97 + \left(19 a + 73\right)\cdot 97^{2} + \left(15 a + 76\right)\cdot 97^{3} + \left(64 a + 83\right)\cdot 97^{4} +O(97^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 87 a + 43 + \left(68 a + 24\right)\cdot 97 + \left(84 a + 7\right)\cdot 97^{2} + \left(13 a + 19\right)\cdot 97^{3} + \left(85 a + 6\right)\cdot 97^{4} +O(97^{5})\) 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$$(1,2)(3,4)(5,6)$$-3$
$15$$2$$(1,2)$$1$
$45$$2$$(1,2)(3,4)$$1$
$40$$3$$(1,2,3)(4,5,6)$$2$
$40$$3$$(1,2,3)$$-1$
$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$$(1,2,3,4,5,6)$$0$
$120$$6$$(1,2,3)(4,5)$$1$

The blue line marks the conjugacy class containing complex conjugation.