Properties

Label 8.749...552.18t157.b.a
Dimension $8$
Group $((C_3^2:Q_8):C_3):C_2$
Conductor $7.493\times 10^{14}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $8$
Group: $((C_3^2:Q_8):C_3):C_2$
Conductor: \(749306075343552\)\(\medspace = 2^{6} \cdot 3^{9} \cdot 29^{6} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 9.3.749306075343552.2
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 9.3.749306075343552.2

Defining polynomial

$f(x)$$=$ \( x^{9} - 3x^{8} - 21x^{7} + 72x^{6} + 60x^{5} - 219x^{4} - 213x^{3} + 309x^{2} - 120x + 3 \) 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:
$r_{ 1 }$ $=$ \( 11 a^{3} + a^{2} + 5 a + 8 + \left(7 a^{3} + 10 a^{2} + 4 a + 9\right)\cdot 13 + \left(6 a^{3} + 6 a^{2} + 8 a + 10\right)\cdot 13^{2} + \left(5 a^{3} + 7 a^{2} + 6 a + 7\right)\cdot 13^{3} + \left(9 a^{3} + a + 6\right)\cdot 13^{4} + \left(10 a^{3} + a^{2} + 9 a + 1\right)\cdot 13^{5} + \left(12 a^{3} + 12 a^{2} + 6 a + 11\right)\cdot 13^{6} + \left(7 a^{3} + 2 a^{2} + 10 a + 11\right)\cdot 13^{7} + \left(a^{3} + 2 a^{2} + 8 a + 10\right)\cdot 13^{8} + \left(9 a^{3} + 9 a^{2} + 9 a + 5\right)\cdot 13^{9} +O(13^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 12 a^{3} + 10 a^{2} + 3 a + 2 + \left(6 a^{3} + 8 a^{2} + 3 a\right)\cdot 13 + \left(4 a^{3} + 5 a^{2}\right)\cdot 13^{2} + \left(9 a^{3} + 12 a^{2} + a + 10\right)\cdot 13^{3} + \left(8 a^{3} + a^{2} + 4 a + 10\right)\cdot 13^{4} + \left(2 a^{3} + 4 a^{2} + 3 a + 8\right)\cdot 13^{5} + \left(6 a^{3} + a^{2} + 3 a + 8\right)\cdot 13^{6} + \left(2 a^{3} + 12 a + 8\right)\cdot 13^{7} + \left(9 a^{3} + 10 a^{2} + 8 a + 2\right)\cdot 13^{8} + \left(4 a^{3} + 7 a^{2} + 10 a + 10\right)\cdot 13^{9} +O(13^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 9 a^{3} + 4 a^{2} + 6 a + 1 + \left(12 a^{3} + 12 a^{2} + 3 a + 11\right)\cdot 13 + \left(9 a^{2} + 12 a + 3\right)\cdot 13^{2} + \left(7 a^{3} + 12 a^{2} + 10 a + 10\right)\cdot 13^{3} + \left(7 a^{3} + 5 a^{2} + 8 a + 10\right)\cdot 13^{4} + \left(4 a^{3} + 3 a^{2} + a + 1\right)\cdot 13^{5} + \left(5 a^{3} + 4 a^{2} + 10\right)\cdot 13^{6} + \left(8 a^{3} + 12 a + 11\right)\cdot 13^{7} + \left(2 a^{2} + 1\right)\cdot 13^{8} + \left(2 a^{2} + 7 a + 11\right)\cdot 13^{9} +O(13^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 12 + 7\cdot 13 + 3\cdot 13^{2} + 11\cdot 13^{3} + 4\cdot 13^{5} + 10\cdot 13^{6} + 5\cdot 13^{8} + 2\cdot 13^{9} +O(13^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 11 a^{3} + 8 a^{2} + 2 a + 3 + \left(12 a^{3} + 12 a^{2} + 11 a + 7\right)\cdot 13 + \left(8 a^{3} + 8 a^{2} + a + 12\right)\cdot 13^{2} + \left(4 a^{3} + 2 a^{2} + 12 a + 11\right)\cdot 13^{3} + \left(10 a^{3} + 4 a^{2} + 2\right)\cdot 13^{4} + \left(5 a^{2} + 3 a\right)\cdot 13^{5} + \left(10 a^{3} + 2 a^{2} + 2 a + 6\right)\cdot 13^{6} + \left(4 a^{3} + 5 a^{2} + 4\right)\cdot 13^{7} + \left(a^{3} + 8 a^{2} + 4 a + 1\right)\cdot 13^{8} + \left(11 a^{3} + 5 a^{2} + 11 a + 6\right)\cdot 13^{9} +O(13^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 6 a^{3} + a^{2} + 10 a + 2 + \left(12 a^{3} + 11 a^{2} + 10 a + 7\right)\cdot 13 + \left(8 a^{3} + 9 a^{2} + 10 a + 10\right)\cdot 13^{2} + \left(11 a^{3} + a^{2} + 6 a + 2\right)\cdot 13^{3} + \left(3 a^{3} + 8 a^{2} + 10 a + 7\right)\cdot 13^{4} + \left(10 a^{3} + 10 a^{2} + 6 a + 5\right)\cdot 13^{5} + \left(a^{3} + 9 a + 5\right)\cdot 13^{6} + \left(11 a^{3} + 7 a^{2} + 12 a + 7\right)\cdot 13^{7} + \left(4 a^{3} + 6 a^{2} + 2 a + 7\right)\cdot 13^{8} + \left(3 a^{3} + 2 a^{2} + 4 a + 2\right)\cdot 13^{9} +O(13^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 7 a^{2} + 5 a + 9 + \left(6 a^{3} + 5 a^{2} + 7 a + 5\right)\cdot 13 + \left(9 a^{3} + 12 a^{2} + 7 a + 12\right)\cdot 13^{2} + \left(a^{3} + 3 a^{2} + a\right)\cdot 13^{3} + \left(5 a^{3} + 11 a^{2} + 5 a + 10\right)\cdot 13^{4} + \left(10 a^{2} + 8 a + 7\right)\cdot 13^{5} + \left(6 a^{3} + 8 a^{2} + 9 a + 3\right)\cdot 13^{6} + \left(11 a^{3} + 2 a^{2} + 3 a + 10\right)\cdot 13^{7} + \left(5 a^{3} + 2 a^{2} + 3\right)\cdot 13^{8} + \left(12 a^{2} + 5 a + 10\right)\cdot 13^{9} +O(13^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 7 a^{3} + 11 a^{2} + 12 a + 4 + \left(2 a^{3} + 8 a^{2} + 11 a + 6\right)\cdot 13 + \left(2 a^{3} + 8 a^{2} + 4 a + 9\right)\cdot 13^{2} + \left(7 a^{3} + 7 a^{2} + 2\right)\cdot 13^{3} + \left(10 a^{2} + 5 a + 2\right)\cdot 13^{4} + \left(12 a^{3} + 11 a^{2} + 2 a + 7\right)\cdot 13^{5} + \left(9 a^{3} + 3 a^{2} + 12 a\right)\cdot 13^{6} + \left(10 a^{2} + 8 a + 2\right)\cdot 13^{7} + \left(11 a^{3} + 9 a^{2} + 12 a + 6\right)\cdot 13^{8} + \left(12 a^{3} + 3 a^{2} + 10 a + 12\right)\cdot 13^{9} +O(13^{10})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 9 a^{3} + 10 a^{2} + 9 a + 1 + \left(3 a^{3} + 8 a^{2} + 12 a + 10\right)\cdot 13 + \left(10 a^{3} + 2 a^{2} + 5 a + 1\right)\cdot 13^{2} + \left(4 a^{3} + 3 a^{2} + 12 a + 7\right)\cdot 13^{3} + \left(6 a^{3} + 9 a^{2} + 2 a\right)\cdot 13^{4} + \left(10 a^{3} + 4 a^{2} + 4 a + 2\right)\cdot 13^{5} + \left(12 a^{3} + 5 a^{2} + 8 a + 9\right)\cdot 13^{6} + \left(4 a^{3} + 10 a^{2} + 4 a + 7\right)\cdot 13^{7} + \left(4 a^{3} + 10 a^{2} + 12\right)\cdot 13^{8} + \left(10 a^{3} + 8 a^{2} + 6 a + 3\right)\cdot 13^{9} +O(13^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.