Properties

Label 18.107...000.36t1758.a.a
Dimension $18$
Group $S_4\wr C_2$
Conductor $1.073\times 10^{28}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $18$
Group: $S_4\wr C_2$
Conductor: \(107\!\cdots\!000\)\(\medspace = 2^{24} \cdot 5^{9} \cdot 41^{9}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 8.4.11027360000.1
Galois orbit size: $1$
Smallest permutation container: 36T1758
Parity: even
Determinant: 1.205.2t1.a.a
Projective image: $S_4\wr C_2$
Projective stem field: Galois closure of 8.4.11027360000.1

Defining polynomial

$f(x)$$=$ \( x^{8} - 2x^{7} + 6x^{6} - 2x^{5} + 26x^{4} - 24x^{3} - 24x^{2} + 16x + 4 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: \( x^{4} + 3x^{2} + 16x + 3 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 13 + 10\cdot 31 + 16\cdot 31^{2} + 21\cdot 31^{3} + 16\cdot 31^{4} + 26\cdot 31^{5} + 27\cdot 31^{6} + 7\cdot 31^{7} + 15\cdot 31^{8} + 26\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 16 a^{3} + 16 a^{2} + 26 a + 21 + \left(16 a^{3} + 13 a^{2} + 21 a + 4\right)\cdot 31 + \left(23 a^{3} + 9 a^{2} + 2 a + 8\right)\cdot 31^{2} + \left(25 a^{3} + 2 a^{2} + 12 a + 18\right)\cdot 31^{3} + \left(12 a^{3} + 7 a^{2} + 12 a + 24\right)\cdot 31^{4} + \left(26 a^{3} + 9 a^{2} + 11 a + 9\right)\cdot 31^{5} + \left(10 a^{3} + 14 a^{2} + 16 a + 17\right)\cdot 31^{6} + \left(7 a^{3} + a^{2} + 7 a + 13\right)\cdot 31^{7} + \left(24 a^{3} + 10 a\right)\cdot 31^{8} + \left(14 a^{3} + 21 a^{2} + 13 a + 25\right)\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 2 a^{3} + 15 a^{2} + 4 a + 19 + \left(17 a^{3} + 8 a^{2} + 14 a\right)\cdot 31 + \left(5 a^{3} + 3 a^{2} + 30 a\right)\cdot 31^{2} + \left(21 a^{3} + 12 a^{2} + 10 a + 13\right)\cdot 31^{3} + \left(16 a^{3} + 8 a^{2} + 23 a + 18\right)\cdot 31^{4} + \left(22 a^{3} + 30 a^{2} + 10 a + 30\right)\cdot 31^{5} + \left(22 a^{3} + 15 a^{2} + 9 a + 30\right)\cdot 31^{6} + \left(23 a^{3} + 8 a^{2} + 24\right)\cdot 31^{7} + \left(21 a^{3} + 7 a^{2} + 16\right)\cdot 31^{8} + \left(23 a^{3} + a^{2} + 20 a + 15\right)\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 29 a^{3} + 16 a^{2} + 27 a + 19 + \left(13 a^{3} + 22 a^{2} + 16 a + 30\right)\cdot 31 + \left(25 a^{3} + 27 a^{2} + 11\right)\cdot 31^{2} + \left(9 a^{3} + 18 a^{2} + 20 a + 26\right)\cdot 31^{3} + \left(14 a^{3} + 22 a^{2} + 7 a + 26\right)\cdot 31^{4} + \left(8 a^{3} + 20 a + 18\right)\cdot 31^{5} + \left(8 a^{3} + 15 a^{2} + 21 a + 26\right)\cdot 31^{6} + \left(7 a^{3} + 22 a^{2} + 30 a + 18\right)\cdot 31^{7} + \left(9 a^{3} + 23 a^{2} + 30 a + 30\right)\cdot 31^{8} + \left(7 a^{3} + 29 a^{2} + 10 a\right)\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 25 a^{3} + 28 a^{2} + 25 a + 23 + \left(a^{3} + 6 a^{2} + 7 a + 19\right)\cdot 31 + \left(29 a^{3} + 19 a^{2} + a + 11\right)\cdot 31^{2} + \left(13 a^{3} + 22 a^{2} + 24 a + 15\right)\cdot 31^{3} + \left(4 a^{3} + 16 a^{2} + 26 a\right)\cdot 31^{4} + \left(8 a^{3} + 14 a^{2} + 3 a\right)\cdot 31^{5} + \left(13 a^{3} + 24 a^{2} + 9 a + 15\right)\cdot 31^{6} + \left(16 a^{3} + 10 a^{2} + 14 a + 12\right)\cdot 31^{7} + \left(18 a^{3} + 23 a^{2} + 5 a + 13\right)\cdot 31^{8} + \left(12 a^{3} + 27 a^{2} + 6 a + 24\right)\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 12 a^{3} + 9 a^{2} + 27 a + 9 + \left(28 a^{3} + 4 a^{2} + 26 a + 9\right)\cdot 31 + \left(21 a^{3} + 28 a^{2} + 17 a + 1\right)\cdot 31^{2} + \left(7 a^{3} + 29 a^{2} + 30 a + 29\right)\cdot 31^{3} + \left(3 a^{3} + 24 a^{2} + 12 a + 13\right)\cdot 31^{4} + \left(9 a^{3} + 21 a^{2} + 16 a + 22\right)\cdot 31^{5} + \left(9 a^{3} + 9 a^{2} + 6 a + 22\right)\cdot 31^{6} + \left(16 a^{3} + 11 a^{2} + 27 a + 11\right)\cdot 31^{7} + \left(16 a^{3} + 24 a^{2} + 7 a + 6\right)\cdot 31^{8} + \left(2 a^{3} + 27 a^{2} + 10 a + 12\right)\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 18 + 7\cdot 31 + 9\cdot 31^{2} + 2\cdot 31^{4} + 30\cdot 31^{5} + 17\cdot 31^{6} + 7\cdot 31^{7} + 14\cdot 31^{8} + 10\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 9 a^{3} + 9 a^{2} + 15 a + 4 + \left(15 a^{3} + 6 a^{2} + 5 a + 10\right)\cdot 31 + \left(18 a^{3} + 5 a^{2} + 9 a + 3\right)\cdot 31^{2} + \left(14 a^{3} + 7 a^{2} + 26 a\right)\cdot 31^{3} + \left(10 a^{3} + 13 a^{2} + 9 a + 21\right)\cdot 31^{4} + \left(18 a^{3} + 16 a^{2} + 30 a + 16\right)\cdot 31^{5} + \left(28 a^{3} + 13 a^{2} + 29 a + 27\right)\cdot 31^{6} + \left(21 a^{3} + 7 a^{2} + 12 a + 26\right)\cdot 31^{7} + \left(2 a^{3} + 14 a^{2} + 7 a + 26\right)\cdot 31^{8} + \left(a^{3} + 16 a^{2} + a + 8\right)\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.