Properties

Label 8.925...409.24t1539.a.b
Dimension $8$
Group $S_3 \wr C_3 $
Conductor $9.255\times 10^{13}$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $8$
Group: $S_3 \wr C_3 $
Conductor: \(92551076400409\)\(\medspace = 11^{4} \cdot 43^{6} \)
Artin stem field: Galois closure of 9.3.69534993539.1
Galois orbit size: $2$
Smallest permutation container: 24T1539
Parity: even
Determinant: 1.43.3t1.a.b
Projective image: $S_3\wr C_3$
Projective stem field: Galois closure of 9.3.69534993539.1

Defining polynomial

$f(x)$$=$ \( x^{9} - 3x^{7} - 4x^{6} + 3x^{5} + 8x^{4} - 10x^{3} - 4x^{2} + 9x + 4 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: \( x^{3} + 6x + 35 \) Copy content Toggle raw display

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.