Properties

Label 12.127...016.18t206.a.a
Dimension $12$
Group $S_3 \wr C_3 $
Conductor $1.279\times 10^{16}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $12$
Group: $S_3 \wr C_3 $
Conductor: \(12786520101630016\)\(\medspace = 2^{6} \cdot 7^{10} \cdot 29^{4}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 9.5.10699470656.1
Galois orbit size: $1$
Smallest permutation container: 18T206
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $C_3^3:C_2^2.C_6$
Projective stem field: Galois closure of 9.5.10699470656.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 83 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 83 }$: \( x^{3} + 3x + 81 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( a^{2} + 67 a + 61 + \left(32 a^{2} + 82 a + 34\right)\cdot 83 + \left(35 a^{2} + 11 a + 33\right)\cdot 83^{2} + \left(66 a^{2} + 63 a + 7\right)\cdot 83^{3} + \left(52 a^{2} + 73 a + 47\right)\cdot 83^{4} + \left(11 a^{2} + 10 a + 3\right)\cdot 83^{5} + \left(33 a^{2} + 42 a + 7\right)\cdot 83^{6} + \left(80 a^{2} + 2 a + 60\right)\cdot 83^{7} + \left(20 a^{2} + 41 a + 81\right)\cdot 83^{8} + \left(32 a^{2} + 29 a + 57\right)\cdot 83^{9} +O(83^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 14 a^{2} + 77 a + 75 + \left(82 a^{2} + 62 a + 32\right)\cdot 83 + \left(2 a^{2} + 25 a + 7\right)\cdot 83^{2} + \left(78 a^{2} + 27 a + 10\right)\cdot 83^{3} + \left(66 a^{2} + 48 a + 49\right)\cdot 83^{4} + \left(61 a^{2} + 33 a + 66\right)\cdot 83^{5} + \left(60 a^{2} + 26 a + 6\right)\cdot 83^{6} + \left(2 a^{2} + 3 a + 58\right)\cdot 83^{7} + \left(62 a^{2} + 34 a + 56\right)\cdot 83^{8} + \left(37 a^{2} + 77 a + 74\right)\cdot 83^{9} +O(83^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 61 a^{2} + 11 a + 15 + \left(39 a^{2} + 60 a + 50\right)\cdot 83 + \left(50 a^{2} + 45 a + 63\right)\cdot 83^{2} + \left(43 a + 41\right)\cdot 83^{3} + \left(66 a^{2} + 4 a + 73\right)\cdot 83^{4} + \left(21 a^{2} + 82 a + 23\right)\cdot 83^{5} + \left(62 a^{2} + 63 a + 65\right)\cdot 83^{6} + \left(82 a^{2} + 73 a + 64\right)\cdot 83^{7} + \left(31 a^{2} + 50 a + 20\right)\cdot 83^{8} + \left(60 a^{2} + 39 a + 31\right)\cdot 83^{9} +O(83^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 21 a^{2} + 5 a + 18 + \left(11 a^{2} + 23 a + 76\right)\cdot 83 + \left(80 a^{2} + 25 a + 39\right)\cdot 83^{2} + \left(15 a^{2} + 59 a + 72\right)\cdot 83^{3} + \left(47 a^{2} + 4 a + 35\right)\cdot 83^{4} + \left(49 a^{2} + 73 a + 79\right)\cdot 83^{5} + \left(70 a^{2} + 59 a + 81\right)\cdot 83^{6} + \left(2 a^{2} + 6 a + 70\right)\cdot 83^{7} + \left(30 a^{2} + 74 a + 16\right)\cdot 83^{8} + \left(73 a^{2} + 13 a + 57\right)\cdot 83^{9} +O(83^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 51 + 49\cdot 83 + 32\cdot 83^{2} + 29\cdot 83^{3} + 48\cdot 83^{4} + 17\cdot 83^{5} + 72\cdot 83^{6} + 39\cdot 83^{7} + 21\cdot 83^{8} + 10\cdot 83^{9} +O(83^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 72 a^{2} + 71 a + 25 + \left(42 a^{2} + 64 a + 37\right)\cdot 83 + \left(71 a + 2\right)\cdot 83^{2} + \left(19 a^{2} + 39 a + 58\right)\cdot 83^{3} + \left(75 a^{2} + 57 a + 65\right)\cdot 83^{4} + \left(2 a^{2} + 82 a + 31\right)\cdot 83^{5} + \left(23 a^{2} + 9 a + 14\right)\cdot 83^{6} + \left(63 a^{2} + 76 a + 13\right)\cdot 83^{7} + \left(33 a^{2} + 9 a\right)\cdot 83^{8} + \left(19 a^{2} + 10 a + 38\right)\cdot 83^{9} +O(83^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 36 + 53\cdot 83 + 19\cdot 83^{2} + 80\cdot 83^{3} + 10\cdot 83^{4} + 59\cdot 83^{5} + 34\cdot 83^{6} + 38\cdot 83^{7} + 16\cdot 83^{8} + 48\cdot 83^{9} +O(83^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 12 + 47\cdot 83 + 55\cdot 83^{2} + 40\cdot 83^{3} + 38\cdot 83^{4} + 70\cdot 83^{5} + 82\cdot 83^{6} + 65\cdot 83^{7} + 44\cdot 83^{8} + 46\cdot 83^{9} +O(83^{10})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 80 a^{2} + 18 a + 41 + \left(40 a^{2} + 38 a + 33\right)\cdot 83 + \left(79 a^{2} + 68 a + 77\right)\cdot 83^{2} + \left(68 a^{2} + 15 a + 74\right)\cdot 83^{3} + \left(23 a^{2} + 60 a + 45\right)\cdot 83^{4} + \left(18 a^{2} + 49 a + 62\right)\cdot 83^{5} + \left(82 a^{2} + 46 a + 49\right)\cdot 83^{6} + \left(16 a^{2} + 3 a + 3\right)\cdot 83^{7} + \left(70 a^{2} + 39 a + 73\right)\cdot 83^{8} + \left(25 a^{2} + 78 a + 50\right)\cdot 83^{9} +O(83^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 9 }$ Character value
$1$$1$$()$$12$
$9$$2$$(5,7)$$4$
$27$$2$$(1,3)(2,6)(5,7)$$0$
$27$$2$$(3,4)(5,7)$$0$
$6$$3$$(2,6,9)$$0$
$8$$3$$(1,3,4)(2,6,9)(5,7,8)$$3$
$12$$3$$(1,3,4)(2,6,9)$$-3$
$36$$3$$(1,5,2)(3,7,6)(4,8,9)$$0$
$36$$3$$(1,2,5)(3,6,7)(4,9,8)$$0$
$18$$6$$(2,6,9)(5,7)$$-2$
$18$$6$$(1,3,4)(5,7)$$-2$
$36$$6$$(1,3,4)(2,6,9)(5,7)$$1$
$54$$6$$(2,6,9)(3,4)(5,7)$$0$
$108$$6$$(1,5,6,3,7,2)(4,8,9)$$0$
$108$$6$$(1,2,7,3,6,5)(4,9,8)$$0$
$72$$9$$(1,5,2,3,7,6,4,8,9)$$0$
$72$$9$$(1,2,7,4,9,5,3,6,8)$$0$

The blue line marks the conjugacy class containing complex conjugation.