Properties

Label 10.333...329.30t164.d.a
Dimension $10$
Group $S_6$
Conductor $3.336\times 10^{16}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $10$
Group: $S_6$
Conductor: \(33364831591822329\)\(\medspace = 3^{14} \cdot 17^{8} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.0.20295603.1
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 6.0.20295603.1

Defining polynomial

$f(x)$$=$ \( x^{6} - x^{5} - 2x^{4} + x^{3} + 14x^{2} - 17x + 6 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 193 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 193 }$: \( x^{2} + 192x + 5 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 67 + 139\cdot 193 + 134\cdot 193^{2} + 101\cdot 193^{3} + 74\cdot 193^{4} +O(193^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 98 + 189\cdot 193 + 100\cdot 193^{2} + 118\cdot 193^{3} + 21\cdot 193^{4} +O(193^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 176 a + 155 + \left(141 a + 115\right)\cdot 193 + \left(138 a + 43\right)\cdot 193^{2} + \left(172 a + 146\right)\cdot 193^{3} + \left(105 a + 112\right)\cdot 193^{4} +O(193^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 98 a + 12 + \left(186 a + 175\right)\cdot 193 + \left(140 a + 55\right)\cdot 193^{2} + \left(98 a + 37\right)\cdot 193^{3} + \left(59 a + 85\right)\cdot 193^{4} +O(193^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 95 a + 110 + \left(6 a + 70\right)\cdot 193 + \left(52 a + 10\right)\cdot 193^{2} + \left(94 a + 188\right)\cdot 193^{3} + \left(133 a + 45\right)\cdot 193^{4} +O(193^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 17 a + 138 + \left(51 a + 81\right)\cdot 193 + \left(54 a + 40\right)\cdot 193^{2} + \left(20 a + 180\right)\cdot 193^{3} + \left(87 a + 45\right)\cdot 193^{4} +O(193^{5})\) 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.