Properties

Label 12.912...776.18t218.a.a
Dimension $12$
Group $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$
Conductor $9.121\times 10^{15}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $12$
Group: $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$
Conductor: \(9121115669451776\)\(\medspace = 2^{10} \cdot 389^{5}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 9.5.5861899530496.1
Galois orbit size: $1$
Smallest permutation container: 18T218
Parity: even
Determinant: 1.389.2t1.a.a
Projective image: $C_3^3.S_4$
Projective stem field: Galois closure of 9.5.5861899530496.1

Defining polynomial

$f(x)$$=$ \( x^{9} - 3x^{8} - 2x^{7} + 12x^{6} - 12x^{5} - 6x^{4} + 16x^{3} - 4x^{2} - 5x + 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$: \( x^{3} + 2x + 68 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 57 a^{2} + 31 a + 61 + \left(67 a^{2} + 66 a + 23\right)\cdot 73 + \left(47 a^{2} + 21 a + 26\right)\cdot 73^{2} + \left(50 a^{2} + 42 a + 4\right)\cdot 73^{3} + \left(22 a^{2} + 50 a + 9\right)\cdot 73^{4} + \left(67 a^{2} + 19 a + 64\right)\cdot 73^{5} + \left(10 a^{2} + 48 a + 13\right)\cdot 73^{6} + \left(46 a^{2} + 20 a + 71\right)\cdot 73^{7} + \left(50 a^{2} + 49 a + 29\right)\cdot 73^{8} + \left(55 a^{2} + 57 a + 11\right)\cdot 73^{9} +O(73^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 37 a^{2} + 12 a + 46 + \left(29 a^{2} + 47 a + 40\right)\cdot 73 + \left(62 a^{2} + 58 a + 31\right)\cdot 73^{2} + \left(26 a^{2} + 32 a + 6\right)\cdot 73^{3} + \left(45 a^{2} + 40 a + 44\right)\cdot 73^{4} + \left(36 a^{2} + 69 a + 34\right)\cdot 73^{5} + \left(35 a^{2} + 52 a + 51\right)\cdot 73^{6} + \left(47 a^{2} + 50 a + 68\right)\cdot 73^{7} + \left(4 a^{2} + 43 a + 18\right)\cdot 73^{8} + \left(64 a^{2} + 41 a + 50\right)\cdot 73^{9} +O(73^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 50 a^{2} + 22 a + 3 + \left(20 a^{2} + 69 a + 34\right)\cdot 73 + \left(58 a^{2} + 24 a + 64\right)\cdot 73^{2} + \left(33 a^{2} + 11 a + 54\right)\cdot 73^{3} + \left(46 a^{2} + 25 a + 40\right)\cdot 73^{4} + \left(57 a^{2} + 40 a + 2\right)\cdot 73^{5} + \left(49 a^{2} + 38 a + 17\right)\cdot 73^{6} + \left(52 a^{2} + 31\right)\cdot 73^{7} + \left(25 a^{2} + 3 a + 45\right)\cdot 73^{8} + \left(38 a^{2} + 4 a + 12\right)\cdot 73^{9} +O(73^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 39 a^{2} + 20 a + 37 + \left(57 a^{2} + 10 a + 34\right)\cdot 73 + \left(39 a^{2} + 26 a + 15\right)\cdot 73^{2} + \left(61 a^{2} + 19 a + 43\right)\cdot 73^{3} + \left(3 a^{2} + 70 a + 32\right)\cdot 73^{4} + \left(21 a^{2} + 12 a + 2\right)\cdot 73^{5} + \left(12 a^{2} + 59 a + 40\right)\cdot 73^{6} + \left(47 a^{2} + 51 a + 72\right)\cdot 73^{7} + \left(69 a^{2} + 20 a + 30\right)\cdot 73^{8} + \left(51 a^{2} + 11 a + 6\right)\cdot 73^{9} +O(73^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 40 a^{2} + 18 a + 50 + \left(52 a^{2} + 16 a + 22\right)\cdot 73 + \left(24 a^{2} + 52 a + 54\right)\cdot 73^{2} + \left(36 a^{2} + 15 a + 67\right)\cdot 73^{3} + \left(14 a^{2} + 7 a + 2\right)\cdot 73^{4} + \left(26 a^{2} + 54 a + 45\right)\cdot 73^{5} + \left(15 a^{2} + 16 a + 24\right)\cdot 73^{6} + \left(46 a^{2} + 29 a + 18\right)\cdot 73^{7} + \left(44 a^{2} + 52 a + 72\right)\cdot 73^{8} + \left(17 a^{2} + 49 a + 36\right)\cdot 73^{9} +O(73^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 56 a^{2} + 33 a + 21 + \left(72 a^{2} + 60 a + 16\right)\cdot 73 + \left(62 a^{2} + 68 a + 27\right)\cdot 73^{2} + \left(2 a^{2} + 45 a + 23\right)\cdot 73^{3} + \left(12 a^{2} + 40 a + 29\right)\cdot 73^{4} + \left(62 a^{2} + 51 a + 25\right)\cdot 73^{5} + \left(7 a^{2} + 17 a + 31\right)\cdot 73^{6} + \left(47 a^{2} + 43 a + 23\right)\cdot 73^{7} + \left(2 a^{2} + 17 a + 28\right)\cdot 73^{8} + \left(17 a^{2} + 19 a + 23\right)\cdot 73^{9} +O(73^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 20 a^{2} + 72 a + 46 + \left(14 a^{2} + 69 a + 35\right)\cdot 73 + \left(39 a^{2} + 15 a + 68\right)\cdot 73^{2} + \left(12 a^{2} + 6 a + 11\right)\cdot 73^{3} + \left(37 a^{2} + 70 a + 14\right)\cdot 73^{4} + \left(68 a^{2} + 30 a + 58\right)\cdot 73^{5} + \left(39 a^{2} + 21 a + 49\right)\cdot 73^{6} + \left(47 a^{2} + 59 a + 72\right)\cdot 73^{7} + \left(71 a^{2} + 46 a + 22\right)\cdot 73^{8} + \left(25 a^{2} + 33 a + 35\right)\cdot 73^{9} +O(73^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 69 a^{2} + 43 a + 40 + \left(63 a^{2} + 9 a + 13\right)\cdot 73 + \left(58 a^{2} + 35 a + 51\right)\cdot 73^{2} + \left(9 a^{2} + 24 a + 56\right)\cdot 73^{3} + \left(13 a^{2} + 25 a + 49\right)\cdot 73^{4} + \left(10 a^{2} + 22 a + 23\right)\cdot 73^{5} + \left(22 a^{2} + 3 a + 9\right)\cdot 73^{6} + \left(52 a^{2} + 66 a + 2\right)\cdot 73^{7} + \left(23 a^{2} + 49 a + 20\right)\cdot 73^{8} + \left(64 a^{2} + 54 a + 26\right)\cdot 73^{9} +O(73^{10})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 70 a^{2} + 41 a + 64 + \left(58 a^{2} + 15 a + 70\right)\cdot 73 + \left(43 a^{2} + 61 a + 25\right)\cdot 73^{2} + \left(57 a^{2} + 20 a + 23\right)\cdot 73^{3} + \left(23 a^{2} + 35 a + 69\right)\cdot 73^{4} + \left(15 a^{2} + 63 a + 35\right)\cdot 73^{5} + \left(25 a^{2} + 33 a + 54\right)\cdot 73^{6} + \left(51 a^{2} + 43 a + 4\right)\cdot 73^{7} + \left(71 a^{2} + 8 a + 23\right)\cdot 73^{8} + \left(29 a^{2} + 20 a + 16\right)\cdot 73^{9} +O(73^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.