Properties

Label 27.334...376.28t165.a.a
Dimension $27$
Group $\mathrm{P}\Gamma\mathrm{L}(2,8)$
Conductor $3.350\times 10^{44}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $27$
Group: $\mathrm{P}\Gamma\mathrm{L}(2,8)$
Conductor: \(334\!\cdots\!376\)\(\medspace = 2^{30} \cdot 7^{42}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 9.1.72313663744.1
Galois orbit size: $1$
Smallest permutation container: 28T165
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $\PSL(2,8).C_3$
Projective stem field: Galois closure of 9.1.72313663744.1

Defining polynomial

$f(x)$$=$ \( x^{9} - x^{8} - 4x^{7} + 28x^{3} + 26x^{2} + 9x + 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 103 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 103 }$: \( x^{3} + 2x + 98 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 34 + 65\cdot 103 + 25\cdot 103^{2} + 85\cdot 103^{3} + 33\cdot 103^{4} + 84\cdot 103^{5} + 15\cdot 103^{6} + 87\cdot 103^{7} + 61\cdot 103^{8} + 33\cdot 103^{9} +O(103^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 59 + 67\cdot 103 + 95\cdot 103^{2} + 85\cdot 103^{3} + 47\cdot 103^{4} + 64\cdot 103^{5} + 6\cdot 103^{6} + 6\cdot 103^{7} + 43\cdot 103^{8} + 25\cdot 103^{9} +O(103^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 75 + 23\cdot 103 + 67\cdot 103^{2} + 48\cdot 103^{3} + 68\cdot 103^{4} + 12\cdot 103^{5} + 14\cdot 103^{6} + 21\cdot 103^{7} + 17\cdot 103^{8} + 52\cdot 103^{9} +O(103^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 32 a + 51 + \left(23 a^{2} + 30 a + 61\right)\cdot 103 + \left(30 a^{2} + 85 a + 43\right)\cdot 103^{2} + \left(13 a^{2} + 42 a + 80\right)\cdot 103^{3} + \left(3 a^{2} + 37 a + 27\right)\cdot 103^{4} + \left(82 a^{2} + 27 a + 2\right)\cdot 103^{5} + \left(96 a^{2} + 52 a + 65\right)\cdot 103^{6} + \left(16 a^{2} + 53 a + 45\right)\cdot 103^{7} + \left(17 a^{2} + 44 a + 50\right)\cdot 103^{8} + \left(67 a^{2} + 18 a + 15\right)\cdot 103^{9} +O(103^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 23 a^{2} + 96 a + 13 + \left(51 a^{2} + 62 a + 99\right)\cdot 103 + \left(72 a^{2} + 58 a + 99\right)\cdot 103^{2} + \left(60 a^{2} + 58 a + 74\right)\cdot 103^{3} + \left(101 a^{2} + 60 a + 21\right)\cdot 103^{4} + \left(72 a^{2} + 73 a + 93\right)\cdot 103^{5} + \left(39 a^{2} + 50 a + 91\right)\cdot 103^{6} + \left(100 a^{2} + 101 a + 53\right)\cdot 103^{7} + \left(31 a^{2} + 17 a + 1\right)\cdot 103^{8} + \left(87 a^{2} + 48 a + 8\right)\cdot 103^{9} +O(103^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 24 a^{2} + 51 a + 97 + \left(47 a^{2} + 45 a + 82\right)\cdot 103 + \left(95 a^{2} + 35 a + 26\right)\cdot 103^{2} + \left(81 a + 71\right)\cdot 103^{3} + \left(9 a^{2} + 77 a + 75\right)\cdot 103^{4} + \left(5 a^{2} + 87 a + 25\right)\cdot 103^{5} + \left(66 a^{2} + 76 a + 71\right)\cdot 103^{6} + \left(13 a^{2} + 35 a + 25\right)\cdot 103^{7} + \left(50 a^{2} + 50 a + 67\right)\cdot 103^{8} + \left(66 a^{2} + 51 a + 22\right)\cdot 103^{9} +O(103^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 34 a^{2} + 10 a + 76 + \left(93 a^{2} + 86 a + 75\right)\cdot 103 + \left(92 a^{2} + 13 a + 57\right)\cdot 103^{2} + \left(41 a^{2} + 83 a + 91\right)\cdot 103^{3} + \left(39 a^{2} + 79 a + 81\right)\cdot 103^{4} + \left(45 a^{2} + 87 a + 10\right)\cdot 103^{5} + \left(31 a^{2} + 29 a + 25\right)\cdot 103^{6} + \left(94 a^{2} + 44 a + 30\right)\cdot 103^{7} + \left(91 a^{2} + 76 a + 54\right)\cdot 103^{8} + \left(95 a^{2} + 100 a + 27\right)\cdot 103^{9} +O(103^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 45 a^{2} + 42 a + 22 + \left(65 a^{2} + 74 a + 4\right)\cdot 103 + \left(17 a^{2} + 53 a + 26\right)\cdot 103^{2} + \left(60 a^{2} + 41 a + 47\right)\cdot 103^{3} + \left(54 a^{2} + 48 a + 33\right)\cdot 103^{4} + \left(52 a^{2} + 30 a + 20\right)\cdot 103^{5} + \left(5 a^{2} + 99 a + 59\right)\cdot 103^{6} + \left(98 a^{2} + 22 a + 69\right)\cdot 103^{7} + \left(63 a^{2} + 79 a + 85\right)\cdot 103^{8} + \left(43 a^{2} + 53 a + 60\right)\cdot 103^{9} +O(103^{10})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 80 a^{2} + 78 a + 89 + \left(28 a^{2} + 9 a + 34\right)\cdot 103 + \left(62 a + 72\right)\cdot 103^{2} + \left(29 a^{2} + a + 32\right)\cdot 103^{3} + \left(101 a^{2} + 5 a + 21\right)\cdot 103^{4} + \left(50 a^{2} + 2 a + 98\right)\cdot 103^{5} + \left(69 a^{2} + 62\right)\cdot 103^{6} + \left(88 a^{2} + 51 a + 72\right)\cdot 103^{7} + \left(53 a^{2} + 40 a + 30\right)\cdot 103^{8} + \left(51 a^{2} + 36 a + 63\right)\cdot 103^{9} +O(103^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 9 }$ Character value
$1$$1$$()$$27$
$63$$2$$(1,3)(2,8)(4,7)(5,9)$$3$
$56$$3$$(1,2,8)(3,6,9)(4,5,7)$$0$
$84$$3$$(2,6,3)(4,5,8)$$0$
$84$$3$$(2,3,6)(4,8,5)$$0$
$252$$6$$(1,2,9,3,8,5)(4,7)$$0$
$252$$6$$(1,5,8,3,9,2)(4,7)$$0$
$216$$7$$(1,3,4,8,6,2,7)$$-1$
$168$$9$$(1,3,7,4,5,8,2,6,9)$$0$
$168$$9$$(1,7,5,2,9,3,4,8,6)$$0$
$168$$9$$(1,3,8,6,9,7,4,2,5)$$0$

The blue line marks the conjugacy class containing complex conjugation.