Properties

Label 8.253...401.12t177.b
Dimension $8$
Group $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$
Conductor $2.531\times 10^{13}$
Indicator $1$

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Basic invariants

Dimension:$8$
Group:$(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$
Conductor:\(25311454040401\)\(\medspace = 2243^{4} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 9.1.25311454040401.1
Galois orbit size: $1$
Smallest permutation container: 12T177
Parity: even
Projective image: $C_3^3:S_4$
Projective field: Galois closure of 9.1.25311454040401.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 97 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 97 }$: \( x^{3} + 9x + 92 \) Copy content Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( a + 17 + \left(12 a^{2} + 37 a + 81\right)\cdot 97 + \left(92 a^{2} + 21 a + 64\right)\cdot 97^{2} + \left(91 a^{2} + 50 a + 10\right)\cdot 97^{3} + \left(85 a^{2} + 67 a + 55\right)\cdot 97^{4} + \left(26 a^{2} + 93 a + 9\right)\cdot 97^{5} + \left(65 a^{2} + 96 a + 66\right)\cdot 97^{6} + \left(84 a^{2} + 45 a + 78\right)\cdot 97^{7} + \left(71 a^{2} + 45 a + 91\right)\cdot 97^{8} + \left(76 a^{2} + 38 a + 79\right)\cdot 97^{9} +O(97^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 13 a^{2} + 75 a + 95 + \left(91 a^{2} + 70 a + 70\right)\cdot 97 + \left(18 a^{2} + 61 a + 13\right)\cdot 97^{2} + \left(20 a^{2} + 31 a + 65\right)\cdot 97^{3} + \left(58 a^{2} + 61 a + 82\right)\cdot 97^{4} + \left(65 a^{2} + 46 a + 47\right)\cdot 97^{5} + \left(57 a^{2} + 24 a + 20\right)\cdot 97^{6} + \left(28 a^{2} + 35 a + 33\right)\cdot 97^{7} + \left(a^{2} + 27 a + 56\right)\cdot 97^{8} + \left(14 a^{2} + 81 a + 91\right)\cdot 97^{9} +O(97^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 29 + 52\cdot 97 + 14\cdot 97^{2} + 7\cdot 97^{3} + 92\cdot 97^{4} + 91\cdot 97^{5} + 38\cdot 97^{6} + 80\cdot 97^{7} + 24\cdot 97^{8} + 20\cdot 97^{9} +O(97^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 88 a^{2} + 48 a + 52 + \left(79 a^{2} + 3 a + 38\right)\cdot 97 + \left(43 a^{2} + 78 a + 91\right)\cdot 97^{2} + \left(31 a^{2} + 18 a + 71\right)\cdot 97^{3} + \left(87 a^{2} + 2 a + 13\right)\cdot 97^{4} + \left(14 a^{2} + 27 a + 46\right)\cdot 97^{5} + \left(5 a^{2} + 71 a + 79\right)\cdot 97^{6} + \left(56 a^{2} + 45 a + 32\right)\cdot 97^{7} + \left(37 a^{2} + 46 a + 84\right)\cdot 97^{8} + \left(56 a^{2} + 81 a + 14\right)\cdot 97^{9} +O(97^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 57 + 49\cdot 97 + 31\cdot 97^{2} + 91\cdot 97^{3} + 18\cdot 97^{4} + 97^{5} + 25\cdot 97^{6} + 95\cdot 97^{7} + 80\cdot 97^{8} + 70\cdot 97^{9} +O(97^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 47 a^{2} + 48 a + \left(74 a^{2} + 37 a + 6\right)\cdot 97 + \left(29 a^{2} + 46 a + 7\right)\cdot 97^{2} + \left(85 a^{2} + 46 a + 7\right)\cdot 97^{3} + \left(62 a^{2} + 14 a + 61\right)\cdot 97^{4} + \left(80 a^{2} + 93 a + 52\right)\cdot 97^{5} + \left(6 a^{2} + 36 a + 89\right)\cdot 97^{6} + \left(74 a^{2} + 73 a + 43\right)\cdot 97^{7} + \left(41 a^{2} + 11 a + 12\right)\cdot 97^{8} + \left(92 a^{2} + 74 a + 37\right)\cdot 97^{9} +O(97^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 32 + 29\cdot 97 + 89\cdot 97^{2} + 31\cdot 97^{3} + 85\cdot 97^{4} + 6\cdot 97^{5} + 88\cdot 97^{6} + 82\cdot 97^{7} + 74\cdot 97^{8} + 80\cdot 97^{9} +O(97^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 84 a^{2} + 21 a + 36 + \left(90 a^{2} + 86 a + 69\right)\cdot 97 + \left(82 a^{2} + 13 a + 9\right)\cdot 97^{2} + \left(81 a^{2} + 15 a + 47\right)\cdot 97^{3} + \left(49 a^{2} + 65 a + 32\right)\cdot 97^{4} + \left(4 a^{2} + 53 a + 69\right)\cdot 97^{5} + \left(71 a^{2} + 72 a + 3\right)\cdot 97^{6} + \left(80 a^{2} + 15 a + 55\right)\cdot 97^{7} + \left(23 a^{2} + 24 a + 94\right)\cdot 97^{8} + \left(6 a^{2} + 74 a + 44\right)\cdot 97^{9} +O(97^{10})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 59 a^{2} + a + 72 + \left(39 a^{2} + 56 a + 87\right)\cdot 97 + \left(23 a^{2} + 69 a + 65\right)\cdot 97^{2} + \left(77 a^{2} + 31 a + 55\right)\cdot 97^{3} + \left(43 a^{2} + 80 a + 43\right)\cdot 97^{4} + \left(a^{2} + 73 a + 62\right)\cdot 97^{5} + \left(85 a^{2} + 85 a + 73\right)\cdot 97^{6} + \left(63 a^{2} + 74 a + 79\right)\cdot 97^{7} + \left(17 a^{2} + 38 a + 61\right)\cdot 97^{8} + \left(45 a^{2} + 38 a + 44\right)\cdot 97^{9} +O(97^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 9 }$ Character values
$c1$
$1$ $1$ $()$ $8$
$27$ $2$ $(3,5)(4,6)$ $0$
$54$ $2$ $(1,3)(2,5)(4,6)(7,8)$ $0$
$6$ $3$ $(4,9,6)$ $-4$
$8$ $3$ $(1,2,8)(3,5,7)(4,6,9)$ $-1$
$12$ $3$ $(3,7,5)(4,9,6)$ $2$
$72$ $3$ $(1,3,4)(2,5,6)(7,9,8)$ $-1$
$54$ $4$ $(3,4,5,6)(7,9)$ $0$
$54$ $6$ $(1,2)(3,5)(4,6,9)$ $0$
$108$ $6$ $(1,2)(3,4,7,9,5,6)$ $0$
$72$ $9$ $(1,3,4,2,5,6,8,7,9)$ $2$
$72$ $9$ $(1,3,4,8,7,9,2,5,6)$ $-1$
$54$ $12$ $(1,5,2,3)(4,9,6)(7,8)$ $0$
$54$ $12$ $(1,5,2,3)(4,6,9)(7,8)$ $0$
The blue line marks the conjugacy class containing complex conjugation.