Properties

Label 10.768...089.30t164.a.a
Dimension $10$
Group $S_6$
Conductor $7.685\times 10^{27}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $10$
Group: $S_6$
Conductor: \(768\!\cdots\!089\)\(\medspace = 31^{6} \cdot 1433^{6} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.0.44423.1
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 6.0.44423.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 131 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 131 }$: \( x^{2} + 127x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 103 a + 58 + \left(50 a + 127\right)\cdot 131 + \left(115 a + 80\right)\cdot 131^{2} + \left(43 a + 79\right)\cdot 131^{3} + \left(90 a + 68\right)\cdot 131^{4} +O(131^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 80 + 67\cdot 131 + 124\cdot 131^{2} + 83\cdot 131^{3} + 98\cdot 131^{4} +O(131^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 49 a + 120 + \left(41 a + 28\right)\cdot 131 + \left(99 a + 41\right)\cdot 131^{2} + \left(41 a + 38\right)\cdot 131^{3} + \left(127 a + 8\right)\cdot 131^{4} +O(131^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 6 + 58\cdot 131 + 43\cdot 131^{2} + 76\cdot 131^{3} + 10\cdot 131^{4} +O(131^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 82 a + 54 + \left(89 a + 14\right)\cdot 131 + \left(31 a + 4\right)\cdot 131^{2} + \left(89 a + 106\right)\cdot 131^{3} + \left(3 a + 82\right)\cdot 131^{4} +O(131^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 28 a + 77 + \left(80 a + 96\right)\cdot 131 + \left(15 a + 98\right)\cdot 131^{2} + \left(87 a + 8\right)\cdot 131^{3} + \left(40 a + 124\right)\cdot 131^{4} +O(131^{5})\) 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.