Properties

Label 12.218...641.18t206.a.a
Dimension $12$
Group $S_3 \wr C_3 $
Conductor $2.188\times 10^{17}$
Root number $1$
Indicator $1$

Related objects

Downloads

Learn more

Basic invariants

Dimension: $12$
Group: $S_3 \wr C_3 $
Conductor: \(218768667202582641\)\(\medspace = 3^{20} \cdot 89^{4}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 9.5.3831158169.1
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 9.5.3831158169.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: \( x^{3} + 3x + 51 \) Copy content Toggle raw display

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.