Properties

Label 10.428...184.30t164.a.a
Dimension $10$
Group $S_6$
Conductor $4.290\times 10^{22}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $10$
Group: $S_6$
Conductor: \(428\!\cdots\!184\)\(\medspace = 2^{20} \cdot 587^{6} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.2.37568.1
Galois orbit size: $1$
Smallest permutation container: 30T164
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $S_6$
Projective stem field: Galois closure of 6.2.37568.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 167 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 167 }$: \( x^{2} + 166x + 5 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 21 a + 3 + \left(150 a + 82\right)\cdot 167 + \left(89 a + 154\right)\cdot 167^{2} + \left(128 a + 73\right)\cdot 167^{3} + \left(142 a + 128\right)\cdot 167^{4} +O(167^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 85 a + 8 + \left(67 a + 69\right)\cdot 167 + \left(33 a + 122\right)\cdot 167^{2} + \left(97 a + 61\right)\cdot 167^{3} + \left(128 a + 31\right)\cdot 167^{4} +O(167^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 146 a + 24 + \left(16 a + 44\right)\cdot 167 + \left(77 a + 94\right)\cdot 167^{2} + \left(38 a + 112\right)\cdot 167^{3} + \left(24 a + 142\right)\cdot 167^{4} +O(167^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 31 a + 5 + \left(60 a + 29\right)\cdot 167 + \left(104 a + 82\right)\cdot 167^{2} + \left(106 a + 62\right)\cdot 167^{3} + \left(58 a + 8\right)\cdot 167^{4} +O(167^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 82 a + 93 + \left(99 a + 51\right)\cdot 167 + \left(133 a + 88\right)\cdot 167^{2} + \left(69 a + 125\right)\cdot 167^{3} + \left(38 a + 62\right)\cdot 167^{4} +O(167^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 136 a + 36 + \left(106 a + 58\right)\cdot 167 + \left(62 a + 126\right)\cdot 167^{2} + \left(60 a + 64\right)\cdot 167^{3} + \left(108 a + 127\right)\cdot 167^{4} +O(167^{5})\) 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$$1$$()$$10$
$15$$2$$(1,2)(3,4)(5,6)$$2$
$15$$2$$(1,2)$$-2$
$45$$2$$(1,2)(3,4)$$-2$
$40$$3$$(1,2,3)(4,5,6)$$1$
$40$$3$$(1,2,3)$$1$
$90$$4$$(1,2,3,4)(5,6)$$0$
$90$$4$$(1,2,3,4)$$0$
$144$$5$$(1,2,3,4,5)$$0$
$120$$6$$(1,2,3,4,5,6)$$-1$
$120$$6$$(1,2,3)(4,5)$$1$

The blue line marks the conjugacy class containing complex conjugation.