Properties

Label 18.146...552.36t1758.a.a
Dimension $18$
Group $S_4\wr C_2$
Conductor $1.465\times 10^{30}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $18$
Group: $S_4\wr C_2$
Conductor: \(146\!\cdots\!552\)\(\medspace = 2^{59} \cdot 3^{26}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 8.2.764411904.1
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 8.2.764411904.1

Defining polynomial

$f(x)$$=$ \( x^{8} - 4x^{7} + 2x^{6} + 4x^{5} - 7x^{4} - 4x^{3} + 14x^{2} - 8x + 1 \) 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 $\mathbb{Q}_{ 31 }$: \( x^{2} + 29x + 3 \) Copy content Toggle raw display

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.