Properties

Label 6.11284642907.36t1121.a.b
Dimension $6$
Group $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$
Conductor $11284642907$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $6$
Group: $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$
Conductor: \(11284642907\)\(\medspace = 2243^{3} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 9.1.25311454040401.1
Galois orbit size: $2$
Smallest permutation container: 36T1121
Parity: odd
Determinant: 1.2243.2t1.a.a
Projective image: $C_3^3:S_4$
Projective stem field: Galois closure of 9.1.25311454040401.1

Defining polynomial

$f(x)$$=$ \( x^{9} - 2x^{8} + 6x^{7} - 18x^{6} + 41x^{5} - 51x^{4} + 106x^{3} - 116x^{2} + 72x + 16 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 97 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 97 }$: \( x^{3} + 9x + 92 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( a + 17 + \left(12 a^{2} + 37 a + 81\right)\cdot 97 + \left(92 a^{2} + 21 a + 64\right)\cdot 97^{2} + \left(91 a^{2} + 50 a + 10\right)\cdot 97^{3} + \left(85 a^{2} + 67 a + 55\right)\cdot 97^{4} + \left(26 a^{2} + 93 a + 9\right)\cdot 97^{5} + \left(65 a^{2} + 96 a + 66\right)\cdot 97^{6} + \left(84 a^{2} + 45 a + 78\right)\cdot 97^{7} + \left(71 a^{2} + 45 a + 91\right)\cdot 97^{8} + \left(76 a^{2} + 38 a + 79\right)\cdot 97^{9} +O(97^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 13 a^{2} + 75 a + 95 + \left(91 a^{2} + 70 a + 70\right)\cdot 97 + \left(18 a^{2} + 61 a + 13\right)\cdot 97^{2} + \left(20 a^{2} + 31 a + 65\right)\cdot 97^{3} + \left(58 a^{2} + 61 a + 82\right)\cdot 97^{4} + \left(65 a^{2} + 46 a + 47\right)\cdot 97^{5} + \left(57 a^{2} + 24 a + 20\right)\cdot 97^{6} + \left(28 a^{2} + 35 a + 33\right)\cdot 97^{7} + \left(a^{2} + 27 a + 56\right)\cdot 97^{8} + \left(14 a^{2} + 81 a + 91\right)\cdot 97^{9} +O(97^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 29 + 52\cdot 97 + 14\cdot 97^{2} + 7\cdot 97^{3} + 92\cdot 97^{4} + 91\cdot 97^{5} + 38\cdot 97^{6} + 80\cdot 97^{7} + 24\cdot 97^{8} + 20\cdot 97^{9} +O(97^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 88 a^{2} + 48 a + 52 + \left(79 a^{2} + 3 a + 38\right)\cdot 97 + \left(43 a^{2} + 78 a + 91\right)\cdot 97^{2} + \left(31 a^{2} + 18 a + 71\right)\cdot 97^{3} + \left(87 a^{2} + 2 a + 13\right)\cdot 97^{4} + \left(14 a^{2} + 27 a + 46\right)\cdot 97^{5} + \left(5 a^{2} + 71 a + 79\right)\cdot 97^{6} + \left(56 a^{2} + 45 a + 32\right)\cdot 97^{7} + \left(37 a^{2} + 46 a + 84\right)\cdot 97^{8} + \left(56 a^{2} + 81 a + 14\right)\cdot 97^{9} +O(97^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 57 + 49\cdot 97 + 31\cdot 97^{2} + 91\cdot 97^{3} + 18\cdot 97^{4} + 97^{5} + 25\cdot 97^{6} + 95\cdot 97^{7} + 80\cdot 97^{8} + 70\cdot 97^{9} +O(97^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 47 a^{2} + 48 a + \left(74 a^{2} + 37 a + 6\right)\cdot 97 + \left(29 a^{2} + 46 a + 7\right)\cdot 97^{2} + \left(85 a^{2} + 46 a + 7\right)\cdot 97^{3} + \left(62 a^{2} + 14 a + 61\right)\cdot 97^{4} + \left(80 a^{2} + 93 a + 52\right)\cdot 97^{5} + \left(6 a^{2} + 36 a + 89\right)\cdot 97^{6} + \left(74 a^{2} + 73 a + 43\right)\cdot 97^{7} + \left(41 a^{2} + 11 a + 12\right)\cdot 97^{8} + \left(92 a^{2} + 74 a + 37\right)\cdot 97^{9} +O(97^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 32 + 29\cdot 97 + 89\cdot 97^{2} + 31\cdot 97^{3} + 85\cdot 97^{4} + 6\cdot 97^{5} + 88\cdot 97^{6} + 82\cdot 97^{7} + 74\cdot 97^{8} + 80\cdot 97^{9} +O(97^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 84 a^{2} + 21 a + 36 + \left(90 a^{2} + 86 a + 69\right)\cdot 97 + \left(82 a^{2} + 13 a + 9\right)\cdot 97^{2} + \left(81 a^{2} + 15 a + 47\right)\cdot 97^{3} + \left(49 a^{2} + 65 a + 32\right)\cdot 97^{4} + \left(4 a^{2} + 53 a + 69\right)\cdot 97^{5} + \left(71 a^{2} + 72 a + 3\right)\cdot 97^{6} + \left(80 a^{2} + 15 a + 55\right)\cdot 97^{7} + \left(23 a^{2} + 24 a + 94\right)\cdot 97^{8} + \left(6 a^{2} + 74 a + 44\right)\cdot 97^{9} +O(97^{10})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 59 a^{2} + a + 72 + \left(39 a^{2} + 56 a + 87\right)\cdot 97 + \left(23 a^{2} + 69 a + 65\right)\cdot 97^{2} + \left(77 a^{2} + 31 a + 55\right)\cdot 97^{3} + \left(43 a^{2} + 80 a + 43\right)\cdot 97^{4} + \left(a^{2} + 73 a + 62\right)\cdot 97^{5} + \left(85 a^{2} + 85 a + 73\right)\cdot 97^{6} + \left(63 a^{2} + 74 a + 79\right)\cdot 97^{7} + \left(17 a^{2} + 38 a + 61\right)\cdot 97^{8} + \left(45 a^{2} + 38 a + 44\right)\cdot 97^{9} +O(97^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 9 }$ Character value
$1$$1$$()$$6$
$27$$2$$(3,5)(4,6)$$-2$
$54$$2$$(1,3)(2,5)(4,6)(7,8)$$0$
$6$$3$$(4,9,6)$$3$
$8$$3$$(1,2,8)(3,5,7)(4,6,9)$$-3$
$12$$3$$(3,7,5)(4,9,6)$$0$
$72$$3$$(1,3,4)(2,5,6)(7,9,8)$$0$
$54$$4$$(3,4,5,6)(7,9)$$0$
$54$$6$$(1,2)(3,5)(4,6,9)$$1$
$108$$6$$(1,2)(3,4,7,9,5,6)$$0$
$72$$9$$(1,3,4,2,5,6,8,7,9)$$0$
$72$$9$$(1,3,4,8,7,9,2,5,6)$$0$
$54$$12$$(1,5,2,3)(4,9,6)(7,8)$$\zeta_{12}^{3} - 2 \zeta_{12}$
$54$$12$$(1,5,2,3)(4,6,9)(7,8)$$-\zeta_{12}^{3} + 2 \zeta_{12}$

The blue line marks the conjugacy class containing complex conjugation.