Properties

Label 8.17256481947.18t157.a.a
Dimension $8$
Group $((C_3^2:Q_8):C_3):C_2$
Conductor $17256481947$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $8$
Group: $((C_3^2:Q_8):C_3):C_2$
Conductor: \(17256481947\)\(\medspace = 3^{7} \cdot 53^{4} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 9.3.17256481947.1
Galois orbit size: $1$
Smallest permutation container: 18T157
Parity: odd
Determinant: 1.3.2t1.a.a
Projective image: $C_3^2:\GL(2,3)$
Projective stem field: Galois closure of 9.3.17256481947.1

Defining polynomial

$f(x)$$=$ \( x^{9} - x^{8} - 3x^{7} + 3x^{6} - 3x^{5} + 6x^{4} + 9x^{3} - 18x^{2} - 9x + 18 \) 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 $\mathbb{Q}_{ 13 }$: \( x^{4} + 3x^{2} + 12x + 2 \) Copy content Toggle raw display

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

Generators of the action on the roots $r_1, \ldots, r_{ 9 }$

Cycle notation
$(1,3,8,5,4,9,6,7)$
$(1,2,4)(5,6,9)$
$(1,2,9,5)(3,6,8,4)$
$(1,4,2)(3,8,7)(5,6,9)$
$(2,6,8)(3,5,4)$
$(1,7,9)(2,8,6)(3,5,4)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 9 }$ Character value
$1$$1$$()$$8$
$9$$2$$(1,4)(3,9)(5,7)(6,8)$$0$
$36$$2$$(1,5)(2,3)(6,7)$$-2$
$8$$3$$(1,7,9)(2,8,6)(3,5,4)$$-1$
$24$$3$$(1,4,2)(5,9,6)$$2$
$48$$3$$(1,3,5)(2,7,9)(4,8,6)$$-1$
$54$$4$$(1,8,4,6)(3,5,9,7)$$0$
$72$$6$$(1,2,6,5,3,7)(4,8,9)$$1$
$72$$6$$(1,4,6,8,3,9)(2,7)$$0$
$54$$8$$(1,3,8,5,4,9,6,7)$$0$
$54$$8$$(1,9,8,7,4,3,6,5)$$0$

The blue line marks the conjugacy class containing complex conjugation.