Properties

Label 12.162...607.18t218.a.a
Dimension $12$
Group $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$
Conductor $1.622\times 10^{17}$
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: \(162198112382958607\)\(\medspace = 2767^{5}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 9.1.58618761251521.1
Galois orbit size: $1$
Smallest permutation container: 18T218
Parity: odd
Determinant: 1.2767.2t1.a.a
Projective image: $C_3^3.S_4$
Projective stem field: Galois closure of 9.1.58618761251521.1

Defining polynomial

$f(x)$$=$ \( x^{9} - 4x^{8} + 6x^{7} - 31x^{6} + 14x^{5} + 156x^{4} + 117x^{3} + 14x^{2} + 9x - 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: \( x^{3} + 3x + 42 \) Copy content Toggle raw display

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.