Properties

Label 18.545...000.36t1758.a.a
Dimension $18$
Group $S_4\wr C_2$
Conductor $5.459\times 10^{30}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $18$
Group: $S_4\wr C_2$
Conductor: \(545\!\cdots\!000\)\(\medspace = 2^{40} \cdot 3^{26} \cdot 5^{9}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 8.0.955514880.1
Galois orbit size: $1$
Smallest permutation container: 36T1758
Parity: odd
Determinant: 1.20.2t1.a.a
Projective image: $S_4\wr C_2$
Projective stem field: Galois closure of 8.0.955514880.1

Defining polynomial

$f(x)$$=$ \( x^{8} - 4x^{7} + 10x^{6} - 16x^{5} + 28x^{4} - 28x^{3} + 20x^{2} - 8x + 2 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: \( x^{3} + x + 35 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 6 a^{2} + 12 a + 38 + \left(4 a^{2} + 6 a + 7\right)\cdot 41 + \left(20 a + 19\right)\cdot 41^{2} + \left(18 a^{2} + 30 a + 22\right)\cdot 41^{3} + \left(14 a^{2} + 39 a + 8\right)\cdot 41^{4} + \left(15 a^{2} + 22 a + 25\right)\cdot 41^{5} + \left(9 a^{2} + 21 a + 1\right)\cdot 41^{6} + \left(5 a^{2} + 10 a + 11\right)\cdot 41^{7} + \left(17 a^{2} + 34 a + 5\right)\cdot 41^{8} + \left(6 a^{2} + 34 a + 36\right)\cdot 41^{9} +O(41^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 15 a^{2} + 23 a + 3 + \left(14 a^{2} + 35 a + 1\right)\cdot 41 + \left(10 a^{2} + 3 a + 26\right)\cdot 41^{2} + \left(20 a^{2} + 33 a + 37\right)\cdot 41^{3} + \left(31 a^{2} + 11 a + 19\right)\cdot 41^{4} + \left(9 a^{2} + 5 a + 21\right)\cdot 41^{5} + \left(4 a^{2} + 22 a + 25\right)\cdot 41^{6} + \left(21 a^{2} + 31 a + 21\right)\cdot 41^{7} + \left(37 a^{2} + 21 a + 32\right)\cdot 41^{8} + \left(7 a^{2} + 34 a + 9\right)\cdot 41^{9} +O(41^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 10 a^{2} + 3 a + 3 + \left(31 a^{2} + 23 a + 39\right)\cdot 41 + \left(28 a^{2} + 10 a + 7\right)\cdot 41^{2} + \left(21 a^{2} + 27 a + 25\right)\cdot 41^{3} + \left(32 a^{2} + 6 a + 26\right)\cdot 41^{4} + \left(15 a^{2} + 14 a + 33\right)\cdot 41^{5} + \left(35 a^{2} + 31 a + 32\right)\cdot 41^{6} + \left(14 a^{2} + 37 a + 12\right)\cdot 41^{7} + \left(31 a^{2} + 11 a + 36\right)\cdot 41^{8} + \left(6 a^{2} + 2 a + 40\right)\cdot 41^{9} +O(41^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 20 a^{2} + 6 a + 20 + \left(22 a^{2} + 40 a + 6\right)\cdot 41 + \left(30 a^{2} + 16 a + 12\right)\cdot 41^{2} + \left(2 a^{2} + 18 a + 12\right)\cdot 41^{3} + \left(36 a^{2} + 30 a + 9\right)\cdot 41^{4} + \left(15 a^{2} + 12 a + 39\right)\cdot 41^{5} + \left(27 a^{2} + 38 a + 40\right)\cdot 41^{6} + \left(14 a^{2} + 39 a + 30\right)\cdot 41^{7} + \left(27 a^{2} + 25 a + 25\right)\cdot 41^{8} + \left(26 a^{2} + 12 a + 8\right)\cdot 41^{9} +O(41^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 28 a^{2} + 32 a + 15 + \left(29 a^{2} + 27 a + 24\right)\cdot 41 + \left(9 a^{2} + 34 a + 22\right)\cdot 41^{2} + \left(24 a + 24\right)\cdot 41^{3} + \left(18 a + 18\right)\cdot 41^{4} + \left(27 a^{2} + 17 a + 27\right)\cdot 41^{5} + \left(21 a^{2} + 19 a + 23\right)\cdot 41^{6} + \left(40 a^{2} + 19 a + 2\right)\cdot 41^{7} + \left(16 a^{2} + 39 a + 13\right)\cdot 41^{8} + \left(20 a^{2} + 6 a + 36\right)\cdot 41^{9} +O(41^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 13 + 27\cdot 41 + 33\cdot 41^{2} + 8\cdot 41^{3} + 26\cdot 41^{4} + 12\cdot 41^{5} + 13\cdot 41^{6} + 32\cdot 41^{7} + 35\cdot 41^{8} + 13\cdot 41^{9} +O(41^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 3 a^{2} + 6 a + 12 + \left(21 a^{2} + 31 a + 32\right)\cdot 41 + \left(2 a^{2} + 36 a + 17\right)\cdot 41^{2} + \left(19 a^{2} + 29 a + 23\right)\cdot 41^{3} + \left(8 a^{2} + 15 a + 10\right)\cdot 41^{4} + \left(39 a^{2} + 9 a + 8\right)\cdot 41^{5} + \left(24 a^{2} + 31 a + 12\right)\cdot 41^{6} + \left(26 a^{2} + 24 a + 34\right)\cdot 41^{7} + \left(33 a^{2} + 30 a + 37\right)\cdot 41^{8} + \left(13 a^{2} + 31 a + 31\right)\cdot 41^{9} +O(41^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 23 + 25\cdot 41 + 24\cdot 41^{2} + 9\cdot 41^{3} + 3\cdot 41^{4} + 37\cdot 41^{5} + 13\cdot 41^{6} + 18\cdot 41^{7} + 18\cdot 41^{8} + 27\cdot 41^{9} +O(41^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.