Properties

Label 8.16836267547.9t26.a.a
Dimension $8$
Group $((C_3^2:Q_8):C_3):C_2$
Conductor $16836267547$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $8$
Group: $((C_3^2:Q_8):C_3):C_2$
Conductor: \(16836267547\)\(\medspace = 11^{3} \cdot 233^{3} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 9.3.16836267547.1
Galois orbit size: $1$
Smallest permutation container: $((C_3^2:Q_8):C_3):C_2$
Parity: odd
Determinant: 1.2563.2t1.a.a
Projective image: $C_3^2:\GL(2,3)$
Projective stem field: Galois closure of 9.3.16836267547.1

Defining polynomial

$f(x)$$=$ \( x^{9} - 2x^{8} + 2x^{7} - x^{6} + 3x^{5} + 3x^{4} - 14x^{3} - 12x^{2} + x + 2 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$: \( x^{4} + 3x^{2} + 40x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 58 + 50\cdot 61 + 55\cdot 61^{2} + 10\cdot 61^{3} + 60\cdot 61^{4} + 43\cdot 61^{5} + 15\cdot 61^{6} + 59\cdot 61^{7} + 53\cdot 61^{8} + 39\cdot 61^{9} +O(61^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( a^{3} + 58 a^{2} + 10 a + 23 + \left(43 a^{3} + 38 a^{2} + 55 a + 19\right)\cdot 61 + \left(38 a^{3} + 18 a^{2} + a + 17\right)\cdot 61^{2} + \left(15 a^{3} + 45 a^{2} + 12 a + 53\right)\cdot 61^{3} + \left(39 a^{3} + 56 a^{2} + 43 a + 13\right)\cdot 61^{4} + \left(16 a^{3} + 13 a^{2} + 37 a + 28\right)\cdot 61^{5} + \left(46 a^{3} + 8 a^{2} + 14 a + 21\right)\cdot 61^{6} + \left(49 a^{3} + 34 a^{2} + 8 a + 50\right)\cdot 61^{7} + \left(40 a^{3} + 35 a^{2} + 6 a + 22\right)\cdot 61^{8} + \left(8 a^{3} + 55 a^{2} + 30 a + 51\right)\cdot 61^{9} +O(61^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 51 a^{3} + 38 a^{2} + 21 a + 29 + \left(41 a^{3} + 2 a^{2} + 15 a + 51\right)\cdot 61 + \left(47 a^{3} + 44 a^{2} + 47 a + 19\right)\cdot 61^{2} + \left(12 a^{3} + 27 a^{2} + 17 a + 2\right)\cdot 61^{3} + \left(27 a^{3} + a^{2} + 30 a + 27\right)\cdot 61^{4} + \left(39 a^{2} + 38 a + 37\right)\cdot 61^{5} + \left(2 a^{3} + 10 a^{2} + 25 a + 8\right)\cdot 61^{6} + \left(18 a^{3} + 2 a^{2} + 44 a + 57\right)\cdot 61^{7} + \left(42 a^{3} + 58 a^{2} + 31 a + 39\right)\cdot 61^{8} + \left(28 a^{3} + 51 a^{2} + 48 a + 36\right)\cdot 61^{9} +O(61^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 6 a^{3} + 26 a^{2} + 18 a + 55 + \left(22 a^{3} + 10 a^{2} + 58 a + 12\right)\cdot 61 + \left(43 a^{3} + 24 a^{2} + 37 a + 9\right)\cdot 61^{2} + \left(48 a^{3} + 47 a^{2} + 57 a + 46\right)\cdot 61^{3} + \left(22 a^{3} + 27 a^{2} + 44 a + 22\right)\cdot 61^{4} + \left(33 a^{3} + 49 a^{2} + 56 a + 31\right)\cdot 61^{5} + \left(23 a^{3} + 11 a^{2} + 48 a + 40\right)\cdot 61^{6} + \left(54 a^{3} + 22 a^{2} + 10 a + 32\right)\cdot 61^{7} + \left(11 a^{3} + 4 a + 58\right)\cdot 61^{8} + \left(a^{3} + 7 a^{2} + 40 a + 53\right)\cdot 61^{9} +O(61^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 56 a^{3} + 47 a^{2} + 20 a + 31 + \left(49 a^{3} + 37 a^{2} + 36 a + 34\right)\cdot 61 + \left(23 a^{3} + 53 a^{2} + 42 a + 46\right)\cdot 61^{2} + \left(32 a^{3} + 41 a^{2} + 41 a + 5\right)\cdot 61^{3} + \left(12 a^{3} + 14 a^{2} + 24 a\right)\cdot 61^{4} + \left(16 a^{3} + 24 a^{2} + 31 a + 58\right)\cdot 61^{5} + \left(12 a^{3} + 56 a^{2} + 8 a + 12\right)\cdot 61^{6} + \left(9 a^{3} + 50 a^{2} + 55 a + 1\right)\cdot 61^{7} + \left(20 a^{3} + 45 a^{2} + 34 a + 39\right)\cdot 61^{8} + \left(47 a^{3} + 27 a^{2} + 8 a + 35\right)\cdot 61^{9} +O(61^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 49 a^{3} + 52 a^{2} + 54 a + 51 + \left(56 a^{3} + 52 a^{2} + 28 a + 26\right)\cdot 61 + \left(52 a^{3} + 37 a^{2} + 48 a + 15\right)\cdot 61^{2} + \left(47 a^{3} + 60 a^{2} + 30 a + 6\right)\cdot 61^{3} + \left(23 a^{3} + 13 a^{2} + 51 a + 4\right)\cdot 61^{4} + \left(38 a^{3} + 45 a^{2} + 11 a + 26\right)\cdot 61^{5} + \left(29 a^{3} + 55 a^{2} + 48 a + 20\right)\cdot 61^{6} + \left(15 a^{3} + 44 a^{2} + 40 a + 14\right)\cdot 61^{7} + \left(35 a^{3} + 18 a^{2} + 57 a + 44\right)\cdot 61^{8} + \left(13 a^{3} + 18 a^{2} + 17 a + 20\right)\cdot 61^{9} +O(61^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 21 a^{3} + 35 a^{2} + 37 a + 9 + \left(41 a^{3} + 27 a^{2} + 22 a + 13\right)\cdot 61 + \left(43 a^{3} + 21 a^{2} + 24 a + 18\right)\cdot 61^{2} + \left(45 a^{3} + 49 a^{2} + 16\right)\cdot 61^{3} + \left(31 a^{3} + 49 a^{2} + 58 a + 22\right)\cdot 61^{4} + \left(5 a^{3} + 23 a^{2} + 33 a + 14\right)\cdot 61^{5} + \left(44 a^{3} + 47 a^{2} + 33 a + 45\right)\cdot 61^{6} + \left(38 a^{3} + 40 a^{2} + 28 a + 3\right)\cdot 61^{7} + \left(3 a^{3} + 9 a^{2} + 26 a + 58\right)\cdot 61^{8} + \left(10 a^{3} + 57 a^{2} + 25 a + 3\right)\cdot 61^{9} +O(61^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 45 a^{3} + 13 a^{2} + 18 a + 16 + \left(46 a^{3} + 59 a^{2} + 49 a + 32\right)\cdot 61 + \left(4 a^{3} + 49 a^{2} + 52 a + 18\right)\cdot 61^{2} + \left(58 a^{3} + 10 a^{2} + 59 a + 28\right)\cdot 61^{3} + \left(56 a^{3} + 7 a^{2} + 53 a + 40\right)\cdot 61^{4} + \left(39 a^{3} + 42 a^{2} + 28 a + 3\right)\cdot 61^{5} + \left(10 a^{3} + 8 a^{2} + 51 a + 15\right)\cdot 61^{6} + \left(47 a^{3} + 4 a^{2} + 32 a + 33\right)\cdot 61^{7} + \left(21 a^{3} + 46 a^{2} + 50 a + 57\right)\cdot 61^{8} + \left(4 a^{3} + 46 a^{2} + 8 a + 55\right)\cdot 61^{9} +O(61^{10})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 15 a^{3} + 36 a^{2} + 5 a + 35 + \left(3 a^{3} + 14 a^{2} + 39 a + 2\right)\cdot 61 + \left(50 a^{3} + 55 a^{2} + 49 a + 43\right)\cdot 61^{2} + \left(43 a^{3} + 21 a^{2} + 23 a + 13\right)\cdot 61^{3} + \left(29 a^{3} + 11 a^{2} + 59 a + 53\right)\cdot 61^{4} + \left(32 a^{3} + 6 a^{2} + 4 a\right)\cdot 61^{5} + \left(14 a^{3} + 45 a^{2} + 13 a + 3\right)\cdot 61^{6} + \left(11 a^{3} + 44 a^{2} + 23 a + 53\right)\cdot 61^{7} + \left(7 a^{3} + 29 a^{2} + 32 a + 52\right)\cdot 61^{8} + \left(8 a^{3} + 40 a^{2} + 3 a + 6\right)\cdot 61^{9} +O(61^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.