Properties

Label 24.382...000.120.a.a
Dimension $24$
Group $A_5 \wr C_2$
Conductor $3.821\times 10^{123}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $24$
Group: $A_5 \wr C_2$
Conductor: \(382\!\cdots\!000\)\(\medspace = 2^{54} \cdot 5^{12} \cdot 71^{20} \cdot 26449^{14}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 10.10.23300317255158046392320000.1
Galois orbit size: $2$
Smallest permutation container: 120
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $A_5 \wr C_2$
Projective stem field: Galois closure of 10.10.23300317255158046392320000.1

Defining polynomial

$f(x)$$=$ \( x^{10} - 2 x^{9} - 345 x^{8} + 720 x^{7} + 33292 x^{6} - 77348 x^{5} - 795480 x^{4} + 1563624 x^{3} + 4991153 x^{2} - 5404190 x - 10994849 \) 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:
$r_{ 1 }$ $=$ \( 20 a^{2} + 8 a + 1 + \left(6 a^{3} + 20 a^{2} + 25 a + 24\right)\cdot 29 + \left(21 a^{3} + 10 a^{2} + 9 a + 5\right)\cdot 29^{2} + \left(10 a^{3} + 20 a^{2} + 17 a + 28\right)\cdot 29^{3} + \left(8 a^{3} + 18 a^{2} + 5 a + 5\right)\cdot 29^{4} + \left(9 a^{3} + 7 a^{2} + 10 a + 18\right)\cdot 29^{5} + \left(a^{2} + 16 a + 3\right)\cdot 29^{6} + \left(4 a^{3} + 5 a^{2} + 20 a + 14\right)\cdot 29^{7} + \left(27 a^{3} + 2 a + 19\right)\cdot 29^{8} + \left(16 a^{3} + 4 a^{2} + 22 a + 15\right)\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 24 a^{3} + 2 a^{2} + 17 a + 21 + \left(3 a^{3} + 27 a^{2} + 8 a + 10\right)\cdot 29 + \left(3 a^{3} + 3 a^{2} + 25 a + 2\right)\cdot 29^{2} + \left(23 a^{3} + 20 a^{2} + 10 a + 19\right)\cdot 29^{3} + \left(22 a^{3} + 22 a^{2} + 23 a + 10\right)\cdot 29^{4} + \left(23 a^{3} + 14 a^{2} + 23 a + 15\right)\cdot 29^{5} + \left(10 a^{3} + 22 a^{2} + 26 a + 3\right)\cdot 29^{6} + \left(16 a^{3} + 6 a^{2} + 18 a + 17\right)\cdot 29^{7} + \left(6 a^{3} + 4 a^{2} + 6 a + 8\right)\cdot 29^{8} + \left(10 a^{3} + 28 a^{2} + 6 a\right)\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 13 a^{3} + 27 a^{2} + 2 + \left(4 a^{3} + 17 a^{2} + 6 a + 11\right)\cdot 29 + \left(11 a^{3} + 17 a^{2} + 2 a + 8\right)\cdot 29^{2} + \left(13 a^{3} + 10 a^{2} + 4 a + 19\right)\cdot 29^{3} + \left(28 a^{3} + 22 a^{2} + 6 a + 3\right)\cdot 29^{4} + \left(16 a^{3} + 2 a^{2} + 15 a + 20\right)\cdot 29^{5} + \left(8 a^{3} + 8 a^{2} + 8 a + 23\right)\cdot 29^{6} + \left(2 a^{3} + 5 a^{2} + 12 a + 16\right)\cdot 29^{7} + \left(21 a^{3} + 22 a^{2} + 14 a + 9\right)\cdot 29^{8} + \left(17 a^{3} + 23 a^{2} + 10 a + 15\right)\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 19 a^{3} + 22 a^{2} + 12 a + 21 + \left(23 a^{3} + 10 a^{2} + 8 a + 9\right)\cdot 29 + \left(22 a^{3} + 18 a^{2} + 7 a + 24\right)\cdot 29^{2} + \left(16 a^{3} + 14 a^{2} + 22 a + 10\right)\cdot 29^{3} + \left(14 a^{3} + 20 a^{2} + 25 a + 12\right)\cdot 29^{4} + \left(5 a^{3} + 7 a^{2} + 3 a + 26\right)\cdot 29^{5} + \left(24 a^{3} + 11 a^{2} + 10 a + 13\right)\cdot 29^{6} + \left(24 a^{3} + 11 a^{2} + 20 a + 15\right)\cdot 29^{7} + \left(26 a^{3} + 2 a^{2} + a + 18\right)\cdot 29^{8} + \left(14 a^{3} + 2 a^{2} + 6 a + 5\right)\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 26 a^{3} + 18 a^{2} + 9 a + 16 + \left(23 a^{3} + 8 a^{2} + 18 a + 17\right)\cdot 29 + \left(2 a^{3} + 11 a^{2} + 9 a + 2\right)\cdot 29^{2} + \left(17 a^{3} + 12 a^{2} + 14 a + 12\right)\cdot 29^{3} + \left(6 a^{3} + 25 a^{2} + 20 a + 14\right)\cdot 29^{4} + \left(26 a^{3} + 10 a^{2} + 28 a + 23\right)\cdot 29^{5} + \left(24 a^{3} + 8 a^{2} + 22 a + 11\right)\cdot 29^{6} + \left(26 a^{3} + 7 a^{2} + 4 a + 12\right)\cdot 29^{7} + \left(11 a^{3} + 4 a^{2} + 10 a + 26\right)\cdot 29^{8} + \left(8 a^{3} + 28 a^{2} + 19 a + 1\right)\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 23 a^{3} + 21 a^{2} + 10 a + 26 + \left(18 a^{3} + 15 a^{2} + 22 a + 12\right)\cdot 29 + \left(12 a^{3} + 28 a^{2} + 24 a + 19\right)\cdot 29^{2} + \left(2 a^{3} + 28 a^{2} + 2 a + 20\right)\cdot 29^{3} + \left(5 a^{3} + 17 a^{2} + 24 a + 22\right)\cdot 29^{4} + \left(13 a^{3} + 13 a^{2} + 9 a + 26\right)\cdot 29^{5} + \left(18 a^{3} + 11 a^{2} + 2 a + 7\right)\cdot 29^{6} + \left(20 a^{3} + 22 a^{2} + 9 a + 22\right)\cdot 29^{7} + \left(14 a^{3} + 17 a^{2} + 11 a + 3\right)\cdot 29^{8} + \left(22 a^{3} + 25 a^{2} + 6 a + 7\right)\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 5 a^{3} + 27 a^{2} + 12 a + 28 + \left(25 a^{3} + a^{2} + 20 a\right)\cdot 29 + \left(25 a^{3} + 25 a^{2} + 3 a + 11\right)\cdot 29^{2} + \left(5 a^{3} + 8 a^{2} + 18 a + 24\right)\cdot 29^{3} + \left(6 a^{3} + 6 a^{2} + 5 a + 17\right)\cdot 29^{4} + \left(5 a^{3} + 14 a^{2} + 5 a + 1\right)\cdot 29^{5} + \left(18 a^{3} + 6 a^{2} + 2 a + 5\right)\cdot 29^{6} + \left(12 a^{3} + 22 a^{2} + 10 a + 12\right)\cdot 29^{7} + \left(22 a^{3} + 24 a^{2} + 22 a + 26\right)\cdot 29^{8} + \left(18 a^{3} + 22 a + 3\right)\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 3 a^{3} + 15 a^{2} + 6 a + 27 + \left(5 a^{3} + 25 a^{2} + 24 a + 20\right)\cdot 29 + \left(21 a^{3} + 12 a^{2} + 28 a + 12\right)\cdot 29^{2} + \left(7 a^{3} + 5 a^{2} + 8 a + 20\right)\cdot 29^{3} + \left(28 a^{3} + 20 a^{2} + 27 a + 24\right)\cdot 29^{4} + \left(23 a^{3} + 8 a^{2} + 12\right)\cdot 29^{5} + \left(28 a^{3} + 21 a^{2} + 15 a + 18\right)\cdot 29^{6} + \left(4 a^{3} + 23 a + 27\right)\cdot 29^{7} + \left(13 a^{3} + 19 a^{2} + 15 a + 23\right)\cdot 29^{8} + \left(10 a^{3} + 17 a^{2} + 4 a + 15\right)\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 26 a^{3} + 5 a^{2} + 7 a + 22 + \left(9 a^{3} + 28 a^{2} + 12 a + 26\right)\cdot 29 + \left(22 a^{3} + 8 a^{2} + 15 a + 21\right)\cdot 29^{2} + \left(14 a^{3} + 6 a^{2} + 23 a + 20\right)\cdot 29^{3} + \left(7 a^{3} + 17 a^{2} + 9 a + 5\right)\cdot 29^{4} + \left(9 a^{3} + 24 a^{2} + 4 a + 8\right)\cdot 29^{5} + \left(3 a^{3} + 26 a^{2} + 28 a + 5\right)\cdot 29^{6} + \left(13 a^{3} + 16 a^{2} + 16 a + 19\right)\cdot 29^{7} + \left(2 a^{3} + 16 a^{2} + 21 a + 9\right)\cdot 29^{8} + \left(8 a^{3} + 10 a^{2} + 2 a + 18\right)\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 10 }$ $=$ \( 6 a^{3} + 17 a^{2} + 6 a + 12 + \left(24 a^{3} + 17 a^{2} + 28 a + 10\right)\cdot 29 + \left(a^{3} + 7 a^{2} + 17 a + 7\right)\cdot 29^{2} + \left(4 a^{3} + 17 a^{2} + 22 a + 27\right)\cdot 29^{3} + \left(17 a^{3} + 2 a^{2} + 25 a + 26\right)\cdot 29^{4} + \left(11 a^{3} + 11 a^{2} + 13 a + 20\right)\cdot 29^{5} + \left(7 a^{3} + 27 a^{2} + 12 a + 22\right)\cdot 29^{6} + \left(19 a^{3} + 17 a^{2} + 8 a + 16\right)\cdot 29^{7} + \left(27 a^{3} + 4 a^{2} + 9 a + 27\right)\cdot 29^{8} + \left(16 a^{3} + 4 a^{2} + 15 a + 2\right)\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 10 }$ Character value
$1$$1$$()$$24$
$30$$2$$(4,5)(7,9)$$-4$
$60$$2$$(1,4)(2,5)(3,10)(6,9)(7,8)$$0$
$225$$2$$(1,2)(4,5)(6,8)(7,9)$$0$
$40$$3$$(4,5,7)$$3$
$400$$3$$(1,2,8)(4,5,7)$$0$
$900$$4$$(1,7,8,4)(2,10,3,5)(6,9)$$0$
$24$$5$$(4,5,7,9,10)$$4 \zeta_{5}^{3} + 4 \zeta_{5}^{2} + 1$
$24$$5$$(4,7,9,10,5)$$-4 \zeta_{5}^{3} - 4 \zeta_{5}^{2} - 3$
$144$$5$$(1,2,8,6,3)(4,5,7,9,10)$$-2 \zeta_{5}^{3} - 2 \zeta_{5}^{2} - 2$
$144$$5$$(1,8,6,3,2)(4,7,9,10,5)$$2 \zeta_{5}^{3} + 2 \zeta_{5}^{2}$
$288$$5$$(1,2,8,6,3)(4,7,9,10,5)$$-1$
$600$$6$$(1,2)(4,5,7)(6,8)$$-1$
$1200$$6$$(1,7,8,5,2,4)(3,10)(6,9)$$0$
$360$$10$$(1,2)(4,5,7,9,10)(6,8)$$1$
$360$$10$$(1,2)(4,7,9,10,5)(6,8)$$1$
$720$$10$$(1,7,8,5,2,10,3,9,6,4)$$0$
$720$$10$$(1,5,2,10,3,9,6,7,8,4)$$0$
$480$$15$$(1,2,8)(4,5,7,9,10)$$\zeta_{5}^{3} + \zeta_{5}^{2} + 1$
$480$$15$$(1,2,8)(4,7,9,10,5)$$-\zeta_{5}^{3} - \zeta_{5}^{2}$

The blue line marks the conjugacy class containing complex conjugation.