Properties

Label 18.148...064.36t1758.b.a
Dimension $18$
Group $S_4\wr C_2$
Conductor $1.485\times 10^{30}$
Root number $1$
Indicator $1$

Related objects

Downloads

Learn more

Basic invariants

Dimension: $18$
Group: $S_4\wr C_2$
Conductor: \(148\!\cdots\!064\)\(\medspace = 2^{40} \cdot 3^{38}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 8.2.46438023168.3
Galois orbit size: $1$
Smallest permutation container: 36T1758
Parity: odd
Determinant: 1.4.2t1.a.a
Projective image: $S_4\wr C_2$
Projective stem field: Galois closure of 8.2.46438023168.3

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: \( x^{3} + 6x + 35 \) Copy content Toggle raw display

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.