Properties

Label 8.126...603.18t157.a.a
Dimension $8$
Group $((C_3^2:Q_8):C_3):C_2$
Conductor $1.267\times 10^{14}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $8$
Group: $((C_3^2:Q_8):C_3):C_2$
Conductor: \(126746061030603\)\(\medspace = 3^{7} \cdot 7^{4} \cdot 17^{6} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 9.3.126746061030603.1
Galois orbit size: $1$
Smallest permutation container: 18T157
Parity: odd
Determinant: 1.3.2t1.a.a
Projective image: $C_3^2:\GL(2,3)$
Projective stem field: Galois closure of 9.3.126746061030603.1

Defining polynomial

$f(x)$$=$ \( x^{9} - 2x^{8} - 14x^{7} + 52x^{6} - 62x^{5} + 49x^{4} - 101x^{3} + 181x^{2} - 161x + 63 \) 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 $\mathbb{Q}_{ 73 }$: \( x^{4} + 16x^{2} + 56x + 5 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 56 + 6\cdot 73 + 18\cdot 73^{2} + 70\cdot 73^{3} + 26\cdot 73^{4} + 19\cdot 73^{5} + 40\cdot 73^{6} + 68\cdot 73^{7} + 69\cdot 73^{8} + 73^{9} +O(73^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 24 a^{3} + 6 a^{2} + 67 a + 32 + \left(13 a^{3} + 54 a^{2} + 13 a + 31\right)\cdot 73 + \left(55 a^{3} + 66 a^{2} + 69 a + 51\right)\cdot 73^{2} + \left(72 a^{3} + a^{2} + 7 a + 19\right)\cdot 73^{3} + \left(50 a^{3} + 9 a^{2} + 57 a + 63\right)\cdot 73^{4} + \left(72 a^{3} + 59 a^{2} + 9 a + 68\right)\cdot 73^{5} + \left(59 a^{3} + 72 a^{2} + 60 a + 65\right)\cdot 73^{6} + \left(30 a^{3} + 20 a^{2} + 68 a + 1\right)\cdot 73^{7} + \left(69 a^{3} + 3 a^{2} + a + 26\right)\cdot 73^{8} + \left(38 a^{3} + 57 a^{2} + 45 a + 24\right)\cdot 73^{9} +O(73^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 43 a^{3} + 46 a^{2} + 30 a + 9 + \left(16 a^{3} + 11 a^{2} + 57 a + 70\right)\cdot 73 + \left(14 a^{3} + 8 a^{2} + 72 a + 63\right)\cdot 73^{2} + \left(40 a^{3} + 7 a^{2} + 64 a + 33\right)\cdot 73^{3} + \left(64 a^{3} + 57 a^{2} + 62 a + 19\right)\cdot 73^{4} + \left(48 a^{3} + 71 a^{2} + 13 a + 39\right)\cdot 73^{5} + \left(33 a^{3} + 47 a^{2} + 17 a + 24\right)\cdot 73^{6} + \left(28 a^{3} + 47 a^{2} + 15 a + 45\right)\cdot 73^{7} + \left(26 a^{3} + 58 a^{2} + 70 a + 59\right)\cdot 73^{8} + \left(9 a^{3} + 3 a^{2} + 23 a + 63\right)\cdot 73^{9} +O(73^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 59 a^{3} + 28 a^{2} + 38 a + 26 + \left(43 a^{3} + 7 a^{2} + 39 a + 11\right)\cdot 73 + \left(58 a^{3} + 6 a^{2} + 56 a + 13\right)\cdot 73^{2} + \left(24 a^{3} + 56 a^{2} + 37 a + 71\right)\cdot 73^{3} + \left(22 a^{3} + 42 a^{2} + 47 a + 29\right)\cdot 73^{4} + \left(38 a^{3} + 19 a^{2} + 65 a + 53\right)\cdot 73^{5} + \left(20 a^{3} + 8 a^{2} + 45 a + 30\right)\cdot 73^{6} + \left(72 a^{3} + 45 a^{2} + 4 a + 40\right)\cdot 73^{7} + \left(66 a^{3} + 23 a^{2} + 21 a + 24\right)\cdot 73^{8} + \left(31 a^{3} + 43 a^{2} + 45 a + 13\right)\cdot 73^{9} +O(73^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 28 a^{3} + 40 a^{2} + 44 a + 34 + \left(62 a^{3} + 6 a^{2} + 49 a + 32\right)\cdot 73 + \left(9 a^{3} + 5 a^{2} + 60 a + 63\right)\cdot 73^{2} + \left(9 a^{3} + 52 a^{2} + 71 a + 11\right)\cdot 73^{3} + \left(31 a^{3} + 68 a^{2} + 57 a + 71\right)\cdot 73^{4} + \left(4 a^{3} + 62 a^{2} + 63 a + 5\right)\cdot 73^{5} + \left(49 a^{3} + 25 a^{2} + 10 a + 34\right)\cdot 73^{6} + \left(18 a^{3} + a^{2} + 54 a + 64\right)\cdot 73^{7} + \left(64 a^{3} + 33 a^{2} + 22 a + 46\right)\cdot 73^{8} + \left(43 a^{2} + 33 a + 68\right)\cdot 73^{9} +O(73^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 72 a^{3} + 68 a^{2} + 11 a + 62 + \left(43 a^{3} + 7 a^{2} + 64 a\right)\cdot 73 + \left(24 a^{3} + 9 a^{2} + 64 a + 58\right)\cdot 73^{2} + \left(38 a^{3} + 43 a^{2} + 21 a + 71\right)\cdot 73^{3} + \left(12 a^{3} + 42 a^{2} + 4 a + 29\right)\cdot 73^{4} + \left(71 a^{3} + 24 a^{2} + 15 a + 20\right)\cdot 73^{5} + \left(57 a^{3} + 37 a^{2} + 4 a + 62\right)\cdot 73^{6} + \left(66 a^{3} + 35 a^{2} + 51 a + 22\right)\cdot 73^{7} + \left(63 a^{3} + 3 a^{2} + 11 a + 15\right)\cdot 73^{8} + \left(68 a^{3} + 33 a^{2} + 28 a + 67\right)\cdot 73^{9} +O(73^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 46 a^{3} + 15 a^{2} + 56 a + 33 + \left(7 a^{3} + 24 a^{2} + 58 a + 13\right)\cdot 73 + \left(44 a^{3} + 29 a^{2} + 70 a + 28\right)\cdot 73^{2} + \left(26 a^{3} + 56 a^{2} + 3 a + 3\right)\cdot 73^{3} + \left(15 a^{3} + 55 a^{2} + 40 a + 60\right)\cdot 73^{4} + \left(53 a^{3} + 35 a^{2} + 6 a + 5\right)\cdot 73^{5} + \left(52 a^{3} + 71 a^{2} + 70 a + 64\right)\cdot 73^{6} + \left(23 a^{3} + 69 a^{2} + 9 a + 24\right)\cdot 73^{7} + \left(52 a^{3} + 25 a^{2} + 18 a + 11\right)\cdot 73^{8} + \left(26 a^{3} + 25 a^{2} + 23 a + 16\right)\cdot 73^{9} +O(73^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 71 a^{3} + 57 a^{2} + 22 a + 32 + \left(4 a^{3} + 29 a^{2} + 63 a + 19\right)\cdot 73 + \left(29 a^{3} + 29 a^{2} + 18 a + 54\right)\cdot 73^{2} + \left(54 a^{3} + 26 a^{2} + 39 a + 54\right)\cdot 73^{3} + \left(43 a^{3} + 63 a^{2} + 68 a + 71\right)\cdot 73^{4} + \left(5 a^{3} + 18 a^{2} + 59 a + 60\right)\cdot 73^{5} + \left(39 a^{3} + 18 a^{2} + 12 a + 13\right)\cdot 73^{6} + \left(21 a^{3} + 56 a^{2} + 43 a + 42\right)\cdot 73^{7} + \left(37 a^{2} + 36 a + 37\right)\cdot 73^{8} + \left(5 a^{3} + 53 a\right)\cdot 73^{9} +O(73^{10})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 22 a^{3} + 32 a^{2} + 24 a + 10 + \left(26 a^{3} + 4 a^{2} + 18 a + 33\right)\cdot 73 + \left(56 a^{3} + 65 a^{2} + 24 a + 14\right)\cdot 73^{2} + \left(25 a^{3} + 48 a^{2} + 44 a + 28\right)\cdot 73^{3} + \left(51 a^{3} + 25 a^{2} + 26 a + 65\right)\cdot 73^{4} + \left(70 a^{3} + 72 a^{2} + 57 a + 17\right)\cdot 73^{5} + \left(51 a^{3} + 9 a^{2} + 70 a + 29\right)\cdot 73^{6} + \left(29 a^{3} + 15 a^{2} + 44 a + 54\right)\cdot 73^{7} + \left(21 a^{3} + 33 a^{2} + 36 a\right)\cdot 73^{8} + \left(37 a^{3} + 12 a^{2} + 39 a + 36\right)\cdot 73^{9} +O(73^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 9 }$ Character value
$1$$1$$()$$8$
$9$$2$$(1,9)(2,4)(3,6)(7,8)$$0$
$36$$2$$(1,9)(2,3)(4,6)$$-2$
$8$$3$$(1,5,9)(2,8,3)(4,6,7)$$-1$
$24$$3$$(1,3,7)(6,8,9)$$2$
$48$$3$$(1,2,8)(3,4,6)(5,9,7)$$-1$
$54$$4$$(1,7,9,8)(2,6,4,3)$$0$
$72$$6$$(1,4,2,9,6,3)(5,7,8)$$1$
$72$$6$$(1,2)(3,7,9,5,4,8)$$0$
$54$$8$$(2,9,3,8,6,5,7,4)$$0$
$54$$8$$(2,5,3,4,6,9,7,8)$$0$

The blue line marks the conjugacy class containing complex conjugation.