Properties

Label 5.9037905787.12t75.a.a
Dimension $5$
Group $A_5\times C_2$
Conductor $9037905787$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $5$
Group: $A_5\times C_2$
Conductor: \(9037905787\)\(\medspace = 2083^{3}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 10.0.39214470002250643.1
Galois orbit size: $1$
Smallest permutation container: 12T75
Parity: odd
Determinant: 1.2083.2t1.a.a
Projective image: $A_5$
Projective stem field: 5.1.4338889.1

Defining polynomial

$f(x)$$=$\(x^{10} - 20 x^{7} - 9 x^{6} + 26 x^{5} + 100 x^{4} + 90 x^{3} + 281 x^{2} - 117 x + 169\)  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 $\mathbb{Q}_{ 17 }$: \(x^{5} + x + 14\)  Toggle raw display

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

Generators of the action on the roots $r_1, \ldots, r_{ 10 }$

Cycle notation
$(1,9,6,5,8,4)(2,10)(3,7)$
$(1,5)(2,10)(3,7)(4,6)(8,9)$
$(2,3,6)(4,10,7)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 10 }$ Character value
$1$$1$$()$$5$
$1$$2$$(1,5)(2,10)(3,7)(4,6)(8,9)$$-5$
$15$$2$$(1,9)(2,4)(3,7)(5,8)(6,10)$$-1$
$15$$2$$(1,8)(2,6)(4,10)(5,9)$$1$
$20$$3$$(1,6,8)(4,9,5)$$-1$
$12$$5$$(1,3,8,2,6)(4,5,7,9,10)$$0$
$12$$5$$(1,2,3,6,8)(4,9,5,10,7)$$0$
$20$$6$$(1,9,6,5,8,4)(2,10)(3,7)$$1$
$12$$10$$(1,10,3,4,8,5,2,7,6,9)$$0$
$12$$10$$(1,4,2,9,3,5,6,10,8,7)$$0$

The blue line marks the conjugacy class containing complex conjugation.