Properties

Label 6.223...033.12t201.a.a
Dimension $6$
Group $S_4\wr C_2$
Conductor $2.239\times 10^{12}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $6$
Group: $S_4\wr C_2$
Conductor: \(2239112174033\)\(\medspace = 7^{4} \cdot 977^{3} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 8.2.6377879282887.1
Galois orbit size: $1$
Smallest permutation container: 12T201
Parity: even
Determinant: 1.977.2t1.a.a
Projective image: $S_4\wr C_2$
Projective stem field: Galois closure of 8.2.6377879282887.1

Defining polynomial

$f(x)$$=$ \( x^{8} - 2x^{7} - x^{6} + 2x^{5} + 34x^{4} - 33x^{3} - 33x^{2} + 28 \) 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:
$r_{ 1 }$ $=$ \( 2 + 4\cdot 11 + 8\cdot 11^{3} + 11^{4} + 10\cdot 11^{5} + 3\cdot 11^{6} + 2\cdot 11^{8} + 10\cdot 11^{9} +O(11^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 9 a^{2} + 9 a + 8 + \left(7 a^{2} + 10 a + 1\right)\cdot 11 + \left(9 a^{2} + 2 a + 9\right)\cdot 11^{2} + \left(5 a^{2} + 9 a + 8\right)\cdot 11^{3} + \left(8 a^{2} + 9 a + 10\right)\cdot 11^{4} + \left(2 a^{2} + 3 a + 3\right)\cdot 11^{5} + \left(5 a^{2} + 2 a + 9\right)\cdot 11^{6} + \left(10 a^{2} + 5 a + 2\right)\cdot 11^{7} + \left(6 a^{2} + 2 a + 1\right)\cdot 11^{8} + \left(5 a^{2} + 7 a + 4\right)\cdot 11^{9} +O(11^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 2 a^{2} + 4 a + 6 + \left(2 a^{2} + 6 a + 1\right)\cdot 11 + \left(10 a^{2} + 6 a + 6\right)\cdot 11^{2} + \left(7 a^{2} + 5 a\right)\cdot 11^{3} + \left(2 a^{2} + 3\right)\cdot 11^{4} + \left(3 a + 4\right)\cdot 11^{5} + \left(9 a^{2} + 9 a + 3\right)\cdot 11^{6} + \left(10 a^{2} + 5 a + 3\right)\cdot 11^{7} + \left(10 a + 4\right)\cdot 11^{8} + \left(3 a^{2} + 10 a + 4\right)\cdot 11^{9} +O(11^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 7 + 3\cdot 11 + 7\cdot 11^{2} + 4\cdot 11^{3} + 11^{4} + 2\cdot 11^{5} + 10\cdot 11^{6} + 8\cdot 11^{7} + 6\cdot 11^{8} +O(11^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 3 a^{2} + 3 a + 2 + \left(8 a + 10\right)\cdot 11 + \left(a^{2} + 2 a + 9\right)\cdot 11^{2} + \left(6 a^{2} + 3 a + 2\right)\cdot 11^{3} + \left(5 a^{2} + 3\right)\cdot 11^{4} + \left(10 a^{2} + 3 a + 2\right)\cdot 11^{5} + \left(5 a^{2} + 9 a + 8\right)\cdot 11^{6} + \left(5 a^{2} + 5 a\right)\cdot 11^{7} + \left(a^{2} + 9 a + 7\right)\cdot 11^{8} + \left(4 a^{2} + 8 a + 1\right)\cdot 11^{9} +O(11^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 9 a + 7 + \left(a^{2} + 4 a + 3\right)\cdot 11 + \left(2 a^{2} + a + 6\right)\cdot 11^{2} + \left(8 a^{2} + 7 a + 4\right)\cdot 11^{3} + \left(10 a^{2} + 6\right)\cdot 11^{4} + \left(7 a^{2} + 4 a + 3\right)\cdot 11^{5} + \left(7 a^{2} + 10 a + 5\right)\cdot 11^{6} + \left(10 a + 4\right)\cdot 11^{7} + \left(3 a^{2} + 8 a + 3\right)\cdot 11^{8} + \left(2 a^{2} + 3 a + 3\right)\cdot 11^{9} +O(11^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 3 a + 9 + \left(8 a^{2} + 7 a + 5\right)\cdot 11 + \left(9 a^{2} + 10 a + 10\right)\cdot 11^{2} + \left(9 a^{2} + 7 a + 7\right)\cdot 11^{3} + \left(a^{2} + 6 a + 5\right)\cdot 11^{4} + \left(6 a^{2} + 3 a + 7\right)\cdot 11^{5} + \left(4 a^{2} + 5 a + 2\right)\cdot 11^{6} + \left(8 a^{2} + 7 a + 8\right)\cdot 11^{7} + \left(2 a^{2} + 7 a + 8\right)\cdot 11^{8} + \left(7 a^{2} + 8 a + 5\right)\cdot 11^{9} +O(11^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 8 a^{2} + 5 a + 5 + \left(2 a^{2} + 6 a + 2\right)\cdot 11 + \left(8 a + 5\right)\cdot 11^{2} + \left(6 a^{2} + 10 a + 6\right)\cdot 11^{3} + \left(3 a^{2} + 3 a\right)\cdot 11^{4} + \left(5 a^{2} + 4 a + 10\right)\cdot 11^{5} + 7 a\cdot 11^{6} + \left(8 a^{2} + 8 a + 4\right)\cdot 11^{7} + \left(6 a^{2} + 4 a + 10\right)\cdot 11^{8} + \left(10 a^{2} + 4 a + 2\right)\cdot 11^{9} +O(11^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.