Properties

Label 8.514...856.9t27.a.a
Dimension $8$
Group $\PSL(2,8)$
Conductor $5.141\times 10^{14}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $8$
Group: $\PSL(2,8)$
Conductor: \(514147280633856\)\(\medspace = 2^{14} \cdot 3^{22} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 9.1.514147280633856.1
Galois orbit size: $1$
Smallest permutation container: $\PSL(2,8)$
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $\SL(2,8)$
Projective stem field: Galois closure of 9.1.514147280633856.1

Defining polynomial

$f(x)$$=$ \( x^{9} - 12x^{6} - 18x^{5} + 36x^{2} - 27x - 128 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: \( x^{3} + x + 14 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 2 a^{2} + 10 a + 8 + \left(9 a^{2} + 7 a + 1\right)\cdot 17 + \left(10 a^{2} + 11 a + 1\right)\cdot 17^{2} + \left(2 a^{2} + 8 a + 9\right)\cdot 17^{3} + \left(15 a^{2} + 8\right)\cdot 17^{4} + \left(2 a^{2} + 2 a + 1\right)\cdot 17^{5} + \left(4 a^{2} + 12\right)\cdot 17^{6} + \left(10 a^{2} + 10 a + 12\right)\cdot 17^{7} + \left(2 a^{2} + 13 a\right)\cdot 17^{8} + \left(6 a^{2} + 12 a + 6\right)\cdot 17^{9} +O(17^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 16 a^{2} + 8 a + 8 + \left(15 a^{2} + 4 a + 8\right)\cdot 17 + \left(4 a^{2} + 5\right)\cdot 17^{2} + \left(11 a^{2} + 16 a + 15\right)\cdot 17^{3} + \left(12 a^{2} + 16\right)\cdot 17^{4} + \left(6 a^{2} + a + 6\right)\cdot 17^{5} + \left(9 a^{2} + 11 a + 1\right)\cdot 17^{6} + \left(13 a^{2} + 14 a + 15\right)\cdot 17^{7} + \left(10 a^{2} + 2 a + 14\right)\cdot 17^{8} + \left(12 a^{2} + 9 a + 4\right)\cdot 17^{9} +O(17^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 4 a^{2} + 13 a + 10 + \left(5 a^{2} + 13 a + 4\right)\cdot 17 + \left(15 a^{2} + a + 8\right)\cdot 17^{2} + \left(a^{2} + 8 a + 14\right)\cdot 17^{3} + \left(10 a^{2} + 8 a + 16\right)\cdot 17^{4} + \left(16 a^{2} + 7 a + 8\right)\cdot 17^{5} + \left(11 a^{2} + 6 a + 3\right)\cdot 17^{6} + \left(12 a^{2} + 4 a + 2\right)\cdot 17^{7} + \left(4 a^{2} + 3 a + 2\right)\cdot 17^{8} + \left(10 a^{2} + 14\right)\cdot 17^{9} +O(17^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 8 a^{2} + 14 a + 14 + \left(10 a^{2} + 15 a + 4\right)\cdot 17 + \left(11 a^{2} + 7 a + 4\right)\cdot 17^{2} + \left(5 a^{2} + 9 a\right)\cdot 17^{3} + \left(16 a^{2} + 11 a + 8\right)\cdot 17^{4} + \left(13 a^{2} + 15 a\right)\cdot 17^{5} + \left(7 a^{2} + 15 a + 6\right)\cdot 17^{6} + \left(8 a^{2} + 13 a\right)\cdot 17^{7} + \left(14 a^{2} + 13 a + 6\right)\cdot 17^{8} + \left(8 a^{2} + 6 a + 2\right)\cdot 17^{9} +O(17^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( a^{2} + 12 a + 8 + \left(16 a^{2} + 3 a\right)\cdot 17 + \left(4 a^{2} + 12 a + 7\right)\cdot 17^{2} + \left(9 a^{2} + 6 a + 2\right)\cdot 17^{3} + \left(2 a^{2} + 6\right)\cdot 17^{4} + \left(10 a^{2} + 14 a + 10\right)\cdot 17^{5} + \left(12 a + 1\right)\cdot 17^{6} + \left(11 a^{2} + 3 a + 1\right)\cdot 17^{7} + \left(9 a^{2} + 15 a + 11\right)\cdot 17^{8} + 15 a^{2} 17^{9} +O(17^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 12 a^{2} + 5 a + 9 + \left(7 a^{2} + a\right)\cdot 17 + \left(7 a^{2} + 5 a + 16\right)\cdot 17^{2} + \left(11 a^{2} + 8 a + 14\right)\cdot 17^{3} + \left(12 a^{2} + 5 a + 6\right)\cdot 17^{4} + \left(16 a^{2} + 9 a + 16\right)\cdot 17^{5} + \left(3 a^{2} + 12 a + 11\right)\cdot 17^{6} + \left(12 a^{2} + 11 a + 2\right)\cdot 17^{7} + \left(16 a^{2} + 7 a + 10\right)\cdot 17^{8} + \left(5 a^{2} + 8 a + 11\right)\cdot 17^{9} +O(17^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 3 a^{2} + 2 a + 3 + \left(8 a + 1\right)\cdot 17 + \left(16 a^{2} + 16\right)\cdot 17^{2} + \left(2 a^{2} + 14\right)\cdot 17^{3} + \left(6 a^{2} + 11 a + 13\right)\cdot 17^{4} + \left(14 a^{2} + 5 a + 14\right)\cdot 17^{5} + \left(8 a^{2} + 4 a + 3\right)\cdot 17^{6} + \left(11 a^{2} + 12 a + 2\right)\cdot 17^{7} + \left(14 a^{2} + 12 a + 3\right)\cdot 17^{8} + \left(4 a^{2} + 12 a + 5\right)\cdot 17^{9} +O(17^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 10 a^{2} + 12 a + 4 + \left(7 a^{2} + 13 a + 14\right)\cdot 17 + \left(8 a + 13\right)\cdot 17^{2} + \left(8 a + 7\right)\cdot 17^{3} + \left(5 a^{2} + 4 a\right)\cdot 17^{4} + 13 a^{2} 17^{5} + \left(16 a^{2} + 7 a + 12\right)\cdot 17^{6} + \left(11 a^{2} + 5 a + 2\right)\cdot 17^{7} + \left(8 a^{2} + 2\right)\cdot 17^{8} + \left(12 a^{2} + a + 16\right)\cdot 17^{9} +O(17^{10})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 12 a^{2} + 9 a + 4 + \left(12 a^{2} + 16 a + 15\right)\cdot 17 + \left(13 a^{2} + 2 a + 12\right)\cdot 17^{2} + \left(5 a^{2} + 2 a + 5\right)\cdot 17^{3} + \left(4 a^{2} + 8 a + 7\right)\cdot 17^{4} + \left(7 a^{2} + 12 a + 8\right)\cdot 17^{5} + \left(4 a^{2} + 14 a + 15\right)\cdot 17^{6} + \left(10 a^{2} + 8 a + 11\right)\cdot 17^{7} + \left(2 a^{2} + 15 a\right)\cdot 17^{8} + \left(8 a^{2} + 15 a + 7\right)\cdot 17^{9} +O(17^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 9 }$ Character value
$1$$1$$()$$8$
$63$$2$$(1,8)(2,3)(5,6)(7,9)$$0$
$56$$3$$(1,5,2)(3,7,6)(4,8,9)$$-1$
$72$$7$$(1,9,2,6,5,8,3)$$1$
$72$$7$$(1,2,5,3,9,6,8)$$1$
$72$$7$$(1,6,3,2,8,9,5)$$1$
$56$$9$$(1,3,6,9,8,5,4,2,7)$$-1$
$56$$9$$(1,6,8,4,7,3,9,5,2)$$-1$
$56$$9$$(1,8,7,9,2,6,4,3,5)$$-1$

The blue line marks the conjugacy class containing complex conjugation.