Properties

Label 18.417...696.36t1758.a.a
Dimension $18$
Group $S_4\wr C_2$
Conductor $4.179\times 10^{29}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $18$
Group: $S_4\wr C_2$
Conductor: \(417\!\cdots\!696\)\(\medspace = 2^{39} \cdot 97^{9}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 8.4.239251750912.1
Galois orbit size: $1$
Smallest permutation container: 36T1758
Parity: even
Determinant: 1.776.2t1.a.a
Projective image: $S_4\wr C_2$
Projective stem field: Galois closure of 8.4.239251750912.1

Defining polynomial

$f(x)$$=$ \( x^{8} - 6x^{6} - 4x^{5} + 10x^{4} - 12x^{3} - 10x^{2} + 16x + 7 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 71 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 71 }$: \( x^{2} + 69x + 7 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 21 a + 43 + \left(19 a + 23\right)\cdot 71 + \left(5 a + 33\right)\cdot 71^{2} + \left(30 a + 51\right)\cdot 71^{3} + \left(70 a + 19\right)\cdot 71^{4} + \left(61 a + 5\right)\cdot 71^{5} + \left(63 a + 2\right)\cdot 71^{6} + \left(42 a + 60\right)\cdot 71^{7} + \left(55 a + 25\right)\cdot 71^{8} + \left(42 a + 37\right)\cdot 71^{9} +O(71^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 62 + 53\cdot 71 + 70\cdot 71^{2} + 43\cdot 71^{3} + 36\cdot 71^{4} + 7\cdot 71^{5} + 26\cdot 71^{6} + 44\cdot 71^{7} + 8\cdot 71^{8} + 49\cdot 71^{9} +O(71^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 41 a + 31 + \left(13 a + 48\right)\cdot 71 + \left(a + 27\right)\cdot 71^{2} + \left(16 a + 58\right)\cdot 71^{3} + \left(69 a + 11\right)\cdot 71^{4} + \left(36 a + 24\right)\cdot 71^{5} + \left(27 a + 13\right)\cdot 71^{6} + \left(52 a + 69\right)\cdot 71^{7} + \left(56 a + 3\right)\cdot 71^{8} + 19 a\cdot 71^{9} +O(71^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 50 a + 14 + \left(51 a + 41\right)\cdot 71 + \left(65 a + 24\right)\cdot 71^{2} + \left(40 a + 35\right)\cdot 71^{3} + 59\cdot 71^{4} + \left(9 a + 58\right)\cdot 71^{5} + \left(7 a + 67\right)\cdot 71^{6} + \left(28 a + 10\right)\cdot 71^{7} + \left(15 a + 23\right)\cdot 71^{8} + \left(28 a + 67\right)\cdot 71^{9} +O(71^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 17 a + 67 + \left(56 a + 66\right)\cdot 71 + \left(15 a + 46\right)\cdot 71^{2} + \left(22 a + 61\right)\cdot 71^{3} + \left(48 a + 36\right)\cdot 71^{4} + \left(6 a + 10\right)\cdot 71^{5} + \left(66 a + 32\right)\cdot 71^{6} + \left(35 a + 48\right)\cdot 71^{7} + \left(10 a + 39\right)\cdot 71^{8} + 13\cdot 71^{9} +O(71^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 66 + 65\cdot 71 + 41\cdot 71^{2} + 66\cdot 71^{3} + 15\cdot 71^{4} + 31\cdot 71^{5} + 24\cdot 71^{6} + 63\cdot 71^{7} + 21\cdot 71^{8} + 59\cdot 71^{9} +O(71^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 54 a + 30 + \left(14 a + 20\right)\cdot 71 + \left(55 a + 22\right)\cdot 71^{2} + \left(48 a + 19\right)\cdot 71^{3} + \left(22 a + 40\right)\cdot 71^{4} + \left(64 a + 46\right)\cdot 71^{5} + \left(4 a + 15\right)\cdot 71^{6} + \left(35 a + 54\right)\cdot 71^{7} + \left(60 a + 24\right)\cdot 71^{8} + \left(70 a + 3\right)\cdot 71^{9} +O(71^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 30 a + 42 + \left(57 a + 34\right)\cdot 71 + \left(69 a + 16\right)\cdot 71^{2} + \left(54 a + 18\right)\cdot 71^{3} + \left(a + 63\right)\cdot 71^{4} + \left(34 a + 28\right)\cdot 71^{5} + \left(43 a + 31\right)\cdot 71^{6} + \left(18 a + 4\right)\cdot 71^{7} + \left(14 a + 65\right)\cdot 71^{8} + \left(51 a + 53\right)\cdot 71^{9} +O(71^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.