Properties

Label 20.126...992.40t1676.a.a
Dimension $20$
Group $(C_2^4:A_5) : C_2$
Conductor $1.263\times 10^{53}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $20$
Group: $(C_2^4:A_5) : C_2$
Conductor: \(126\!\cdots\!992\)\(\medspace = 2^{40} \cdot 36497^{9}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 10.10.1363999753216.1
Galois orbit size: $1$
Smallest permutation container: 40T1676
Parity: even
Determinant: 1.36497.2t1.a.a
Projective image: $(C_2^4:A_5) : C_2$
Projective stem field: Galois closure of 10.10.1363999753216.1

Defining polynomial

$f(x)$$=$ \( x^{10} - 9x^{8} + 27x^{6} - 31x^{4} + 12x^{2} - 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: \( x^{4} + 2x^{2} + 15x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 16 a^{3} + 26 a^{2} + 3 a + 9 + \left(6 a^{3} + 26 a^{2} + 15 a + 21\right)\cdot 29 + \left(28 a^{3} + 19 a^{2} + 7 a + 19\right)\cdot 29^{2} + \left(17 a^{3} + 7 a^{2} + 22 a + 8\right)\cdot 29^{3} + \left(15 a^{3} + 5 a^{2} + 2 a + 17\right)\cdot 29^{4} + \left(27 a^{3} + 6 a^{2} + 9 a + 26\right)\cdot 29^{5} + \left(26 a^{3} + 15 a^{2} + 17 a + 21\right)\cdot 29^{6} + \left(25 a^{3} + 14 a^{2} + 21\right)\cdot 29^{7} + \left(14 a^{3} + 7 a^{2} + 4 a + 24\right)\cdot 29^{8} + \left(19 a^{3} + 23 a^{2} + 24 a\right)\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 8 + 6\cdot 29 + 25\cdot 29^{2} + 13\cdot 29^{3} + 22\cdot 29^{4} + 14\cdot 29^{5} + 25\cdot 29^{6} + 6\cdot 29^{7} + 7\cdot 29^{8} + 9\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 10 a^{3} + a^{2} + 17 a + 18 + \left(3 a^{3} + 21 a^{2} + 21 a + 15\right)\cdot 29 + \left(20 a^{3} + a^{2} + 26 a + 4\right)\cdot 29^{2} + \left(26 a^{3} + 26 a^{2} + 28 a + 9\right)\cdot 29^{3} + \left(18 a^{3} + 16 a^{2} + 17 a + 15\right)\cdot 29^{4} + \left(20 a^{3} + 17 a^{2} + 14 a + 18\right)\cdot 29^{5} + \left(2 a^{3} + 6 a^{2} + 10 a + 1\right)\cdot 29^{6} + \left(22 a^{3} + 20 a^{2} + 13 a + 6\right)\cdot 29^{7} + \left(18 a^{3} + 20 a^{2} + 16\right)\cdot 29^{8} + \left(23 a^{3} + 4 a^{2} + 7 a + 21\right)\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 9 a^{3} + 23 a^{2} + 21 a + 24 + \left(6 a^{2} + 9 a + 9\right)\cdot 29 + \left(26 a^{3} + 14 a + 23\right)\cdot 29^{2} + \left(28 a^{3} + 17 a^{2} + 14 a + 6\right)\cdot 29^{3} + \left(27 a^{3} + 27 a^{2} + 16 a + 13\right)\cdot 29^{4} + \left(27 a^{3} + 27 a^{2} + 2 a + 8\right)\cdot 29^{5} + \left(24 a^{3} + a^{2} + 6 a + 28\right)\cdot 29^{6} + \left(28 a^{3} + 3 a^{2} + 9 a + 24\right)\cdot 29^{7} + \left(15 a^{3} + 22 a^{2} + 22 a + 18\right)\cdot 29^{8} + \left(6 a^{3} + 19 a + 4\right)\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 17 a^{3} + 4 a^{2} + 28 a + 8 + \left(9 a^{3} + 12 a^{2} + 26 a + 25\right)\cdot 29 + \left(22 a^{3} + 21 a^{2} + 19 a + 24\right)\cdot 29^{2} + \left(15 a^{3} + 16 a^{2} + 7 a + 3\right)\cdot 29^{3} + \left(6 a^{3} + 23 a^{2} + 4 a + 7\right)\cdot 29^{4} + \left(20 a^{3} + 24 a^{2} + 21 a + 5\right)\cdot 29^{5} + \left(4 a^{3} + 19 a^{2} + 21 a + 21\right)\cdot 29^{6} + \left(19 a^{3} + 2 a^{2} + 4 a + 9\right)\cdot 29^{7} + \left(17 a^{3} + 6 a^{2} + 11 a + 14\right)\cdot 29^{8} + \left(7 a^{3} + 27 a^{2} + 11 a + 28\right)\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 13 a^{3} + 3 a^{2} + 26 a + 20 + \left(22 a^{3} + 2 a^{2} + 13 a + 7\right)\cdot 29 + \left(9 a^{2} + 21 a + 9\right)\cdot 29^{2} + \left(11 a^{3} + 21 a^{2} + 6 a + 20\right)\cdot 29^{3} + \left(13 a^{3} + 23 a^{2} + 26 a + 11\right)\cdot 29^{4} + \left(a^{3} + 22 a^{2} + 19 a + 2\right)\cdot 29^{5} + \left(2 a^{3} + 13 a^{2} + 11 a + 7\right)\cdot 29^{6} + \left(3 a^{3} + 14 a^{2} + 28 a + 7\right)\cdot 29^{7} + \left(14 a^{3} + 21 a^{2} + 24 a + 4\right)\cdot 29^{8} + \left(9 a^{3} + 5 a^{2} + 4 a + 28\right)\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 21 + 22\cdot 29 + 3\cdot 29^{2} + 15\cdot 29^{3} + 6\cdot 29^{4} + 14\cdot 29^{5} + 3\cdot 29^{6} + 22\cdot 29^{7} + 21\cdot 29^{8} + 19\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 19 a^{3} + 28 a^{2} + 12 a + 11 + \left(25 a^{3} + 7 a^{2} + 7 a + 13\right)\cdot 29 + \left(8 a^{3} + 27 a^{2} + 2 a + 24\right)\cdot 29^{2} + \left(2 a^{3} + 2 a^{2} + 19\right)\cdot 29^{3} + \left(10 a^{3} + 12 a^{2} + 11 a + 13\right)\cdot 29^{4} + \left(8 a^{3} + 11 a^{2} + 14 a + 10\right)\cdot 29^{5} + \left(26 a^{3} + 22 a^{2} + 18 a + 27\right)\cdot 29^{6} + \left(6 a^{3} + 8 a^{2} + 15 a + 22\right)\cdot 29^{7} + \left(10 a^{3} + 8 a^{2} + 28 a + 12\right)\cdot 29^{8} + \left(5 a^{3} + 24 a^{2} + 21 a + 7\right)\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 20 a^{3} + 6 a^{2} + 8 a + 5 + \left(28 a^{3} + 22 a^{2} + 19 a + 19\right)\cdot 29 + \left(2 a^{3} + 28 a^{2} + 14 a + 5\right)\cdot 29^{2} + \left(11 a^{2} + 14 a + 22\right)\cdot 29^{3} + \left(a^{3} + a^{2} + 12 a + 15\right)\cdot 29^{4} + \left(a^{3} + a^{2} + 26 a + 20\right)\cdot 29^{5} + \left(4 a^{3} + 27 a^{2} + 22 a\right)\cdot 29^{6} + \left(25 a^{2} + 19 a + 4\right)\cdot 29^{7} + \left(13 a^{3} + 6 a^{2} + 6 a + 10\right)\cdot 29^{8} + \left(22 a^{3} + 28 a^{2} + 9 a + 24\right)\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 10 }$ $=$ \( 12 a^{3} + 25 a^{2} + a + 21 + \left(19 a^{3} + 16 a^{2} + 2 a + 3\right)\cdot 29 + \left(6 a^{3} + 7 a^{2} + 9 a + 4\right)\cdot 29^{2} + \left(13 a^{3} + 12 a^{2} + 21 a + 25\right)\cdot 29^{3} + \left(22 a^{3} + 5 a^{2} + 24 a + 21\right)\cdot 29^{4} + \left(8 a^{3} + 4 a^{2} + 7 a + 23\right)\cdot 29^{5} + \left(24 a^{3} + 9 a^{2} + 7 a + 7\right)\cdot 29^{6} + \left(9 a^{3} + 26 a^{2} + 24 a + 19\right)\cdot 29^{7} + \left(11 a^{3} + 22 a^{2} + 17 a + 14\right)\cdot 29^{8} + \left(21 a^{3} + a^{2} + 17 a\right)\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 10 }$ Character value
$1$$1$$()$$20$
$5$$2$$(1,6)(2,7)(3,8)(4,9)$$4$
$10$$2$$(1,6)(5,10)$$-4$
$20$$2$$(1,10)(5,6)$$2$
$60$$2$$(2,5)(3,4)(7,10)(8,9)$$0$
$60$$2$$(1,5)(2,7)(4,9)(6,10)$$2$
$80$$3$$(1,2,5)(6,7,10)$$-1$
$20$$4$$(1,4,6,9)(2,7)(3,8)(5,10)$$-2$
$60$$4$$(1,7,6,2)(3,9,8,4)$$0$
$60$$4$$(1,3,6,8)(2,7)$$-2$
$120$$4$$(1,6)(2,10,7,5)(3,4)(8,9)$$0$
$240$$4$$(2,3,5,4)(7,8,10,9)$$0$
$384$$5$$(1,5,4,2,3)(6,10,9,7,8)$$0$
$80$$6$$(1,6)(2,8,5)(3,10,7)(4,9)$$-1$
$160$$6$$(1,5,2)(3,4)(6,10,7)(8,9)$$-1$
$160$$6$$(1,4,10,6,9,5)(2,7)$$1$
$240$$8$$(1,9,7,8,6,4,2,3)(5,10)$$0$
$160$$12$$(1,9,6,4)(2,10,8,7,5,3)$$1$

The blue line marks the conjugacy class containing complex conjugation.