Properties

Label 18.496...256.36t1758.a.a
Dimension $18$
Group $S_4\wr C_2$
Conductor $4.965\times 10^{29}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $18$
Group: $S_4\wr C_2$
Conductor: \(496\!\cdots\!256\)\(\medspace = 2^{12} \cdot 7^{9} \cdot 113^{9}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 8.2.4565326108.1
Galois orbit size: $1$
Smallest permutation container: 36T1758
Parity: odd
Determinant: 1.791.2t1.a.a
Projective image: $S_4\wr C_2$
Projective stem field: Galois closure of 8.2.4565326108.1

Defining polynomial

$f(x)$$=$ \( x^{8} - 2x^{7} + 3x^{6} - 4x^{5} + 14x^{4} - 13x^{3} + 12x^{2} - 11x + 2 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: \( x^{3} + 2x + 9 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 5 + 9\cdot 11^{2} + 8\cdot 11^{3} + 9\cdot 11^{4} + 7\cdot 11^{5} + 3\cdot 11^{6} + 7\cdot 11^{7} + 10\cdot 11^{8} + 11^{9} +O(11^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 4 a^{2} + 10 a + 4 + \left(9 a^{2} + 10 a + 1\right)\cdot 11 + \left(9 a^{2} + 5 a + 10\right)\cdot 11^{2} + \left(8 a^{2} + 3 a + 8\right)\cdot 11^{3} + \left(3 a^{2} + 5 a + 1\right)\cdot 11^{4} + \left(2 a^{2} + 8 a + 4\right)\cdot 11^{5} + \left(2 a^{2} + 6 a + 5\right)\cdot 11^{6} + \left(5 a^{2} + 7 a + 4\right)\cdot 11^{7} + \left(9 a^{2} + 5\right)\cdot 11^{8} + \left(a^{2} + 3 a + 5\right)\cdot 11^{9} +O(11^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 8 a^{2} + 9 a + 8 + \left(2 a^{2} + 9 a + 4\right)\cdot 11 + \left(6 a^{2} + a + 5\right)\cdot 11^{2} + \left(5 a^{2} + a + 4\right)\cdot 11^{3} + \left(3 a^{2} + a + 8\right)\cdot 11^{4} + \left(10 a^{2} + 5\right)\cdot 11^{5} + \left(9 a^{2} + 4 a + 1\right)\cdot 11^{6} + \left(9 a^{2} + 7 a + 3\right)\cdot 11^{7} + \left(6 a^{2} + 4 a + 5\right)\cdot 11^{8} + \left(8 a^{2} + 6 a + 9\right)\cdot 11^{9} +O(11^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 9 + 7\cdot 11 + 8\cdot 11^{2} + 8\cdot 11^{3} + 10\cdot 11^{4} + 11^{5} + 2\cdot 11^{6} + 8\cdot 11^{7} + 6\cdot 11^{9} +O(11^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 5 a^{2} + 3 a + 9 + \left(5 a^{2} + 5 a + 10\right)\cdot 11 + \left(3 a^{2} + a + 8\right)\cdot 11^{2} + \left(5 a^{2} + 9 a + 7\right)\cdot 11^{3} + \left(7 a^{2} + a + 6\right)\cdot 11^{4} + \left(6 a^{2} + 4 a + 2\right)\cdot 11^{5} + \left(6 a^{2} + 8 a\right)\cdot 11^{6} + \left(5 a^{2} + 9 a + 5\right)\cdot 11^{7} + \left(9 a^{2} + 4 a + 5\right)\cdot 11^{8} + \left(9 a^{2} + 6 a + 1\right)\cdot 11^{9} +O(11^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 2 a^{2} + 9 a + 5 + \left(7 a^{2} + 5 a + 9\right)\cdot 11 + \left(8 a^{2} + 3 a + 4\right)\cdot 11^{2} + \left(7 a^{2} + 9 a + 7\right)\cdot 11^{3} + \left(10 a^{2} + 3 a + 3\right)\cdot 11^{4} + \left(a^{2} + 9 a + 7\right)\cdot 11^{5} + \left(2 a^{2} + 6 a + 1\right)\cdot 11^{6} + \left(4 a + 5\right)\cdot 11^{7} + \left(3 a^{2} + 5 a\right)\cdot 11^{8} + \left(10 a^{2} + a + 2\right)\cdot 11^{9} +O(11^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 10 a^{2} + 6 a + 7 + \left(6 a^{2} + 8 a + 6\right)\cdot 11 + \left(a^{2} + 2 a + 6\right)\cdot 11^{2} + \left(3 a^{2} + 3 a + 8\right)\cdot 11^{3} + \left(10 a^{2} + 2 a + 2\right)\cdot 11^{4} + \left(7 a^{2} + a + 6\right)\cdot 11^{5} + \left(2 a^{2} + 2 a + 6\right)\cdot 11^{6} + \left(5 a^{2} + 10 a\right)\cdot 11^{7} + \left(8 a^{2} + 8 a\right)\cdot 11^{8} + \left(8 a^{2} + 3 a + 6\right)\cdot 11^{9} +O(11^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 4 a^{2} + 7 a + 10 + \left(a^{2} + 3 a + 2\right)\cdot 11 + \left(3 a^{2} + 6 a + 1\right)\cdot 11^{2} + \left(2 a^{2} + 6 a\right)\cdot 11^{3} + \left(8 a^{2} + 7 a\right)\cdot 11^{4} + \left(3 a^{2} + 9 a + 8\right)\cdot 11^{5} + \left(9 a^{2} + 4 a\right)\cdot 11^{6} + \left(6 a^{2} + 4 a + 10\right)\cdot 11^{7} + \left(6 a^{2} + 8 a + 4\right)\cdot 11^{8} + 4 a^{2} 11^{9} +O(11^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.