Properties

Label 18.262...000.36t1758.a.a
Dimension $18$
Group $S_4\wr C_2$
Conductor $2.629\times 10^{30}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $18$
Group: $S_4\wr C_2$
Conductor: \(262\!\cdots\!000\)\(\medspace = 2^{51} \cdot 3^{14} \cdot 5^{12}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 8.2.119439360000.2
Galois orbit size: $1$
Smallest permutation container: 36T1758
Parity: odd
Determinant: 1.8.2t1.b.a
Projective image: $S_4\wr C_2$
Projective stem field: Galois closure of 8.2.119439360000.2

Defining polynomial

$f(x)$$=$ \( x^{8} - 2x^{6} - 12x^{5} + 6x^{4} + 48x^{3} - 32x^{2} + 16 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 89 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 89 }$: \( x^{3} + 3x + 86 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 43 a^{2} + 8 a + 22 + \left(17 a^{2} + 48 a + 34\right)\cdot 89 + \left(23 a^{2} + 49 a + 71\right)\cdot 89^{2} + \left(14 a^{2} + 71 a + 5\right)\cdot 89^{3} + \left(47 a^{2} + 43 a + 54\right)\cdot 89^{4} + \left(36 a^{2} + 32 a + 73\right)\cdot 89^{5} + \left(61 a^{2} + 53 a + 27\right)\cdot 89^{6} + \left(48 a^{2} + 34 a + 65\right)\cdot 89^{7} + \left(59 a + 23\right)\cdot 89^{8} + \left(63 a^{2} + 51 a + 77\right)\cdot 89^{9} +O(89^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 20 + 5\cdot 89 + 60\cdot 89^{2} + 41\cdot 89^{3} + 8\cdot 89^{4} + 86\cdot 89^{5} + 43\cdot 89^{6} + 18\cdot 89^{7} + 8\cdot 89^{8} + 88\cdot 89^{9} +O(89^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 24 a^{2} + 73 a + 73 + \left(67 a^{2} + 36 a + 44\right)\cdot 89 + \left(75 a^{2} + 62 a + 87\right)\cdot 89^{2} + \left(88 a^{2} + 27 a + 65\right)\cdot 89^{3} + \left(80 a^{2} + 47 a + 32\right)\cdot 89^{4} + \left(79 a^{2} + 5 a + 71\right)\cdot 89^{5} + \left(68 a^{2} + 60 a + 42\right)\cdot 89^{6} + \left(69 a^{2} + 18\right)\cdot 89^{7} + \left(61 a^{2} + 71 a + 57\right)\cdot 89^{8} + \left(27 a^{2} + 32 a + 6\right)\cdot 89^{9} +O(89^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 78 + 67\cdot 89 + 63\cdot 89^{2} + 18\cdot 89^{3} + 6\cdot 89^{4} + 79\cdot 89^{5} + 77\cdot 89^{6} + 3\cdot 89^{7} + 11\cdot 89^{8} + 21\cdot 89^{9} +O(89^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 22 a^{2} + 8 a + 69 + \left(4 a^{2} + 4 a + 7\right)\cdot 89 + \left(79 a^{2} + 66 a + 5\right)\cdot 89^{2} + \left(74 a^{2} + 78 a + 38\right)\cdot 89^{3} + \left(49 a^{2} + 86 a + 59\right)\cdot 89^{4} + \left(61 a^{2} + 50 a + 34\right)\cdot 89^{5} + \left(47 a^{2} + 64 a\right)\cdot 89^{6} + \left(59 a^{2} + 53 a + 87\right)\cdot 89^{7} + \left(26 a^{2} + 47 a + 75\right)\cdot 89^{8} + \left(87 a^{2} + 4 a + 36\right)\cdot 89^{9} +O(89^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 21 a^{2} + 57 a + 14 + \left(78 a^{2} + 16 a + 14\right)\cdot 89 + \left(35 a^{2} + 41 a + 35\right)\cdot 89^{2} + \left(9 a^{2} + 16 a + 21\right)\cdot 89^{3} + \left(35 a^{2} + 40 a + 46\right)\cdot 89^{4} + \left(22 a^{2} + 60 a + 48\right)\cdot 89^{5} + \left(3 a^{2} + 86 a + 1\right)\cdot 89^{6} + \left(55 a^{2} + 2 a + 16\right)\cdot 89^{7} + \left(8 a^{2} + 77 a + 77\right)\cdot 89^{8} + \left(41 a^{2} + 87 a + 64\right)\cdot 89^{9} +O(89^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 25 a^{2} + 36 a + 22 + \left(56 a^{2} + 77 a + 59\right)\cdot 89 + \left(73 a^{2} + 48 a + 21\right)\cdot 89^{2} + \left(21 a^{2} + 85 a + 46\right)\cdot 89^{3} + \left(71 a^{2} + 13 a + 29\right)\cdot 89^{4} + \left(67 a^{2} + 78 a + 50\right)\cdot 89^{5} + \left(83 a^{2} + 56 a + 73\right)\cdot 89^{6} + \left(71 a^{2} + 61 a + 49\right)\cdot 89^{7} + \left(16 a^{2} + 60 a + 4\right)\cdot 89^{8} + \left(16 a^{2} + 7 a + 15\right)\cdot 89^{9} +O(89^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 43 a^{2} + 85 a + 58 + \left(43 a^{2} + 83 a + 33\right)\cdot 89 + \left(68 a^{2} + 87 a + 11\right)\cdot 89^{2} + \left(57 a^{2} + 75 a + 29\right)\cdot 89^{3} + \left(71 a^{2} + 34 a + 30\right)\cdot 89^{4} + \left(87 a^{2} + 39 a + 1\right)\cdot 89^{5} + \left(a^{2} + 34 a + 88\right)\cdot 89^{6} + \left(51 a^{2} + 24 a + 7\right)\cdot 89^{7} + \left(63 a^{2} + 40 a + 9\right)\cdot 89^{8} + \left(31 a^{2} + 82 a + 46\right)\cdot 89^{9} +O(89^{10})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 8 }$

Cycle notation
$(1,3,4,5)$
$(1,3)$
$(1,2)(3,6)(4,7)(5,8)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character value
$1$$1$$()$$18$
$6$$2$$(2,7)(6,8)$$-6$
$9$$2$$(1,4)(2,7)(3,5)(6,8)$$2$
$12$$2$$(1,3)$$0$
$24$$2$$(1,2)(3,6)(4,7)(5,8)$$0$
$36$$2$$(1,3)(2,6)$$-2$
$36$$2$$(1,3)(2,7)(6,8)$$0$
$16$$3$$(1,4,5)$$0$
$64$$3$$(1,4,5)(6,7,8)$$0$
$12$$4$$(2,6,7,8)$$0$
$36$$4$$(1,3,4,5)(2,6,7,8)$$-2$
$36$$4$$(1,3,4,5)(2,7)(6,8)$$0$
$72$$4$$(1,2,4,7)(3,6,5,8)$$0$
$72$$4$$(1,3)(2,6,7,8)$$2$
$144$$4$$(1,6,3,2)(4,7)(5,8)$$0$
$48$$6$$(1,5,4)(2,7)(6,8)$$0$
$96$$6$$(1,3)(6,8,7)$$0$
$192$$6$$(1,6,4,7,5,8)(2,3)$$0$
$144$$8$$(1,2,3,6,4,7,5,8)$$0$
$96$$12$$(1,4,5)(2,6,7,8)$$0$

The blue line marks the conjugacy class containing complex conjugation.