Properties

Label 9.160...792.18t274.a.a
Dimension $9$
Group $S_4\wr C_2$
Conductor $1.606\times 10^{20}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $9$
Group: $S_4\wr C_2$
Conductor: \(160\!\cdots\!792\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 673^{6} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 8.2.2461736148492.1
Galois orbit size: $1$
Smallest permutation container: 18T274
Parity: odd
Determinant: 1.3.2t1.a.a
Projective image: $S_4\wr C_2$
Projective stem field: Galois closure of 8.2.2461736148492.1

Defining polynomial

$f(x)$$=$ \( x^{8} - 4x^{7} - 4x^{6} + 16x^{5} + 65x^{4} - 98x^{3} - 196x^{2} + 432 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: \( x^{3} + 2x + 11 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 11 a^{2} + 4 a + \left(12 a^{2} + 10 a + 4\right)\cdot 13 + \left(2 a^{2} + 12\right)\cdot 13^{2} + \left(a^{2} + 4 a + 12\right)\cdot 13^{3} + \left(8 a^{2} + 4 a + 7\right)\cdot 13^{4} + \left(5 a^{2} + 2 a + 2\right)\cdot 13^{5} + \left(10 a^{2} + 3 a + 6\right)\cdot 13^{6} + \left(6 a^{2} + 5 a\right)\cdot 13^{7} + \left(12 a^{2} + 11 a + 10\right)\cdot 13^{8} + \left(3 a^{2} + 11 a + 6\right)\cdot 13^{9} +O(13^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 5 a^{2} + 9 a + 1 + \left(3 a^{2} + 6 a + 12\right)\cdot 13 + \left(7 a^{2} + 11 a + 9\right)\cdot 13^{2} + \left(2 a^{2} + 4 a + 12\right)\cdot 13^{3} + \left(7 a^{2} + 4 a\right)\cdot 13^{4} + \left(12 a^{2} + 4 a + 6\right)\cdot 13^{5} + \left(10 a^{2} + 10 a + 10\right)\cdot 13^{6} + \left(10 a^{2} + 7 a + 8\right)\cdot 13^{7} + \left(a^{2} + 10 a + 8\right)\cdot 13^{8} + \left(2 a^{2} + 3 a + 7\right)\cdot 13^{9} +O(13^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 5 a^{2} + 5 a + 5 + \left(11 a^{2} + 5 a + 6\right)\cdot 13 + \left(7 a^{2} + a + 1\right)\cdot 13^{2} + \left(9 a^{2} + 11 a + 11\right)\cdot 13^{3} + \left(11 a^{2} + 10 a + 12\right)\cdot 13^{4} + \left(4 a^{2} + 11 a + 5\right)\cdot 13^{5} + \left(10 a^{2} + 6 a + 10\right)\cdot 13^{6} + \left(9 a^{2} + 8 a + 8\right)\cdot 13^{7} + \left(10 a^{2} + 11 a + 7\right)\cdot 13^{8} + \left(a^{2} + a + 12\right)\cdot 13^{9} +O(13^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 6 + 3\cdot 13 + 12\cdot 13^{2} + 10\cdot 13^{3} + 12\cdot 13^{4} + 5\cdot 13^{5} + 12\cdot 13^{6} + 3\cdot 13^{7} + 7\cdot 13^{8} + 11\cdot 13^{9} +O(13^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 10 a^{2} + 4 a + 3 + \left(a^{2} + 10 a + 2\right)\cdot 13 + \left(2 a^{2} + 10 a + 11\right)\cdot 13^{2} + \left(2 a^{2} + 10 a + 9\right)\cdot 13^{3} + \left(6 a^{2} + 10 a + 9\right)\cdot 13^{4} + \left(2 a^{2} + 11 a + 2\right)\cdot 13^{5} + \left(5 a^{2} + 2 a + 12\right)\cdot 13^{6} + \left(9 a^{2} + 12 a + 3\right)\cdot 13^{7} + \left(2 a^{2} + 2 a + 1\right)\cdot 13^{8} + \left(7 a^{2} + 12 a + 11\right)\cdot 13^{9} +O(13^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 3 a^{2} + 11 a + 7 + \left(a^{2} + 8 a\right)\cdot 13 + \left(6 a^{2} + 10 a + 4\right)\cdot 13^{2} + \left(6 a^{2} + 6 a + 9\right)\cdot 13^{3} + \left(11 a^{2} + 10 a + 6\right)\cdot 13^{4} + \left(12 a^{2} + 10 a + 6\right)\cdot 13^{5} + \left(3 a^{2} + 4 a + 5\right)\cdot 13^{6} + \left(a^{2} + 2 a\right)\cdot 13^{7} + \left(3 a^{2} + 7 a + 6\right)\cdot 13^{8} + \left(10 a^{2} + 9 a + 5\right)\cdot 13^{9} +O(13^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 5 a^{2} + 6 a + 1 + \left(8 a^{2} + 10 a + 10\right)\cdot 13 + \left(12 a^{2} + 3 a + 12\right)\cdot 13^{2} + \left(3 a^{2} + a + 5\right)\cdot 13^{3} + \left(7 a^{2} + 11 a + 5\right)\cdot 13^{4} + \left(10 a + 7\right)\cdot 13^{5} + \left(11 a^{2} + 10 a + 10\right)\cdot 13^{6} + \left(2 a + 12\right)\cdot 13^{7} + \left(8 a^{2} + 8 a + 3\right)\cdot 13^{8} + \left(12 a + 1\right)\cdot 13^{9} +O(13^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 7 + 13^{2} + 5\cdot 13^{3} + 8\cdot 13^{4} + 13^{5} + 10\cdot 13^{6} + 12\cdot 13^{7} + 6\cdot 13^{8} + 8\cdot 13^{9} +O(13^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.