Properties

Label 10.288...944.30t164.a.a
Dimension $10$
Group $S_6$
Conductor $2.889\times 10^{15}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $10$
Group: $S_6$
Conductor: \(2888816545234944\)\(\medspace = 2^{26} \cdot 3^{16}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 6.2.13436928.5
Galois orbit size: $1$
Smallest permutation container: 30T164
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $S_6$
Projective field: Galois closure of 6.2.13436928.5

Defining polynomial

$f(x)$$=$$ x^{6} - 6 x^{4} - 4 x^{3} + 6 x^{2} - 6 $.

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} + 166 x + 5 $

Roots:
$r_{ 1 }$ $=$ $ 125 + 13\cdot 167 + 71\cdot 167^{2} + 8\cdot 167^{3} + 52\cdot 167^{4} +O\left(167^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 141 + 151\cdot 167 + 21\cdot 167^{2} + 13\cdot 167^{3} + 133\cdot 167^{4} +O\left(167^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 67 a + 128 + \left(164 a + 103\right)\cdot 167 + \left(112 a + 44\right)\cdot 167^{2} + \left(59 a + 111\right)\cdot 167^{3} + \left(a + 25\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 100 a + 28 + \left(2 a + 34\right)\cdot 167 + \left(54 a + 160\right)\cdot 167^{2} + \left(107 a + 57\right)\cdot 167^{3} + \left(165 a + 134\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 131 a + 141 + \left(44 a + 141\right)\cdot 167 + \left(132 a + 57\right)\cdot 167^{2} + \left(58 a + 108\right)\cdot 167^{3} + \left(17 a + 98\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 36 a + 105 + \left(122 a + 55\right)\cdot 167 + \left(34 a + 145\right)\cdot 167^{2} + \left(108 a + 34\right)\cdot 167^{3} + \left(149 a + 57\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$

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.