Properties

Label 4.29e3_229e3.8t44.1c1
Dimension 4
Group $C_2 \wr S_4$
Conductor $ 29^{3} \cdot 229^{3}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$4$
Group:$C_2 \wr S_4$
Conductor:$292887232721= 29^{3} \cdot 229^{3} $
Artin number field: Splitting field of $f= x^{8} - x^{7} - x^{6} + x^{4} - x^{2} + x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $C_2 \wr S_4$
Parity: Even
Determinant: 1.29_229.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 24.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $ x^{2} + 45 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 3 a + 22 + \left(7 a + 33\right)\cdot 47 + \left(32 a + 11\right)\cdot 47^{2} + \left(7 a + 12\right)\cdot 47^{3} + \left(15 a + 16\right)\cdot 47^{4} + \left(36 a + 28\right)\cdot 47^{5} + \left(28 a + 22\right)\cdot 47^{6} + \left(39 a + 43\right)\cdot 47^{7} + \left(42 a + 10\right)\cdot 47^{8} + \left(10 a + 36\right)\cdot 47^{9} + \left(40 a + 18\right)\cdot 47^{10} + \left(29 a + 36\right)\cdot 47^{11} + \left(41 a + 38\right)\cdot 47^{12} + \left(11 a + 29\right)\cdot 47^{13} + \left(33 a + 14\right)\cdot 47^{14} + \left(21 a + 27\right)\cdot 47^{15} + 30 a\cdot 47^{16} + \left(2 a + 16\right)\cdot 47^{17} + \left(37 a + 42\right)\cdot 47^{18} + \left(14 a + 30\right)\cdot 47^{19} + \left(45 a + 4\right)\cdot 47^{20} + \left(26 a + 42\right)\cdot 47^{21} + \left(30 a + 38\right)\cdot 47^{22} + \left(41 a + 14\right)\cdot 47^{23} +O\left(47^{ 24 }\right)$
$r_{ 2 }$ $=$ $ 11 + 26\cdot 47 + 28\cdot 47^{2} + 26\cdot 47^{3} + 28\cdot 47^{4} + 17\cdot 47^{5} + 47^{6} + 18\cdot 47^{7} + 41\cdot 47^{8} + 8\cdot 47^{9} + 32\cdot 47^{10} + 11\cdot 47^{11} + 46\cdot 47^{12} + 26\cdot 47^{14} + 25\cdot 47^{15} + 13\cdot 47^{16} + 38\cdot 47^{17} + 21\cdot 47^{18} + 39\cdot 47^{19} + 23\cdot 47^{20} + 33\cdot 47^{21} + 18\cdot 47^{22} + 36\cdot 47^{23} +O\left(47^{ 24 }\right)$
$r_{ 3 }$ $=$ $ 46 a + 24 + \left(44 a + 43\right)\cdot 47 + \left(34 a + 13\right)\cdot 47^{2} + \left(21 a + 11\right)\cdot 47^{3} + \left(14 a + 32\right)\cdot 47^{4} + \left(35 a + 24\right)\cdot 47^{5} + \left(15 a + 30\right)\cdot 47^{6} + \left(21 a + 1\right)\cdot 47^{7} + \left(32 a + 40\right)\cdot 47^{8} + \left(2 a + 2\right)\cdot 47^{9} + \left(27 a + 45\right)\cdot 47^{10} + \left(36 a + 37\right)\cdot 47^{11} + \left(21 a + 27\right)\cdot 47^{12} + \left(7 a + 16\right)\cdot 47^{13} + \left(33 a + 32\right)\cdot 47^{14} + \left(18 a + 46\right)\cdot 47^{15} + \left(33 a + 2\right)\cdot 47^{16} + \left(4 a + 4\right)\cdot 47^{17} + \left(11 a + 38\right)\cdot 47^{18} + \left(34 a + 14\right)\cdot 47^{19} + \left(41 a + 26\right)\cdot 47^{20} + \left(12 a + 22\right)\cdot 47^{21} + \left(27 a + 12\right)\cdot 47^{22} + \left(26 a + 19\right)\cdot 47^{23} +O\left(47^{ 24 }\right)$
$r_{ 4 }$ $=$ $ 35 a + 21 + \left(6 a + 26\right)\cdot 47 + \left(34 a + 30\right)\cdot 47^{2} + \left(33 a + 42\right)\cdot 47^{3} + \left(43 a + 36\right)\cdot 47^{4} + \left(42 a + 19\right)\cdot 47^{5} + \left(8 a + 19\right)\cdot 47^{6} + \left(25 a + 46\right)\cdot 47^{7} + \left(32 a + 45\right)\cdot 47^{8} + \left(41 a + 22\right)\cdot 47^{9} + \left(2 a + 22\right)\cdot 47^{10} + \left(22 a + 42\right)\cdot 47^{11} + \left(10 a + 41\right)\cdot 47^{12} + 30 a\cdot 47^{13} + \left(27 a + 22\right)\cdot 47^{14} + \left(16 a + 25\right)\cdot 47^{15} + \left(25 a + 43\right)\cdot 47^{16} + 28 a\cdot 47^{17} + \left(25 a + 27\right)\cdot 47^{18} + \left(8 a + 8\right)\cdot 47^{19} + \left(18 a + 26\right)\cdot 47^{20} + \left(27 a + 41\right)\cdot 47^{21} + \left(9 a + 16\right)\cdot 47^{22} + \left(28 a + 33\right)\cdot 47^{23} +O\left(47^{ 24 }\right)$
$r_{ 5 }$ $=$ $ 12 a + 44 + \left(40 a + 4\right)\cdot 47 + \left(12 a + 45\right)\cdot 47^{2} + \left(13 a + 28\right)\cdot 47^{3} + \left(3 a + 43\right)\cdot 47^{4} + \left(4 a + 14\right)\cdot 47^{5} + \left(38 a + 41\right)\cdot 47^{6} + \left(21 a + 40\right)\cdot 47^{7} + \left(14 a + 38\right)\cdot 47^{8} + \left(5 a + 26\right)\cdot 47^{9} + \left(44 a + 33\right)\cdot 47^{10} + \left(24 a + 36\right)\cdot 47^{11} + \left(36 a + 40\right)\cdot 47^{12} + \left(16 a + 3\right)\cdot 47^{13} + 19 a\cdot 47^{14} + \left(30 a + 31\right)\cdot 47^{15} + \left(21 a + 30\right)\cdot 47^{16} + \left(18 a + 32\right)\cdot 47^{17} + \left(21 a + 2\right)\cdot 47^{18} + 38 a\cdot 47^{19} + \left(28 a + 7\right)\cdot 47^{20} + \left(19 a + 31\right)\cdot 47^{21} + \left(37 a + 8\right)\cdot 47^{22} + \left(18 a + 33\right)\cdot 47^{23} +O\left(47^{ 24 }\right)$
$r_{ 6 }$ $=$ $ 44 a + 28 + \left(39 a + 44\right)\cdot 47 + \left(14 a + 21\right)\cdot 47^{2} + \left(39 a + 42\right)\cdot 47^{3} + \left(31 a + 38\right)\cdot 47^{4} + \left(10 a + 38\right)\cdot 47^{5} + \left(18 a + 43\right)\cdot 47^{6} + \left(7 a + 46\right)\cdot 47^{7} + \left(4 a + 9\right)\cdot 47^{8} + \left(36 a + 15\right)\cdot 47^{9} + \left(6 a + 41\right)\cdot 47^{10} + \left(17 a + 8\right)\cdot 47^{11} + \left(5 a + 45\right)\cdot 47^{12} + \left(35 a + 11\right)\cdot 47^{13} + \left(13 a + 22\right)\cdot 47^{14} + \left(25 a + 37\right)\cdot 47^{15} + \left(16 a + 39\right)\cdot 47^{16} + \left(44 a + 37\right)\cdot 47^{17} + \left(9 a + 19\right)\cdot 47^{18} + \left(32 a + 23\right)\cdot 47^{19} + \left(a + 33\right)\cdot 47^{20} + \left(20 a + 3\right)\cdot 47^{21} + \left(16 a + 26\right)\cdot 47^{22} + \left(5 a + 20\right)\cdot 47^{23} +O\left(47^{ 24 }\right)$
$r_{ 7 }$ $=$ $ 17 + 15\cdot 47 + 44\cdot 47^{2} + 3\cdot 47^{3} + 46\cdot 47^{4} + 9\cdot 47^{5} + 2\cdot 47^{6} + 9\cdot 47^{7} + 11\cdot 47^{8} + 5\cdot 47^{9} + 39\cdot 47^{10} + 23\cdot 47^{11} + 6\cdot 47^{12} + 20\cdot 47^{13} + 26\cdot 47^{14} + 37\cdot 47^{15} + 5\cdot 47^{16} + 31\cdot 47^{17} + 27\cdot 47^{18} + 45\cdot 47^{19} + 37\cdot 47^{20} + 6\cdot 47^{21} + 12\cdot 47^{22} + 32\cdot 47^{23} +O\left(47^{ 24 }\right)$
$r_{ 8 }$ $=$ $ a + 22 + \left(2 a + 40\right)\cdot 47 + \left(12 a + 38\right)\cdot 47^{2} + \left(25 a + 19\right)\cdot 47^{3} + \left(32 a + 39\right)\cdot 47^{4} + \left(11 a + 33\right)\cdot 47^{5} + \left(31 a + 26\right)\cdot 47^{6} + \left(25 a + 28\right)\cdot 47^{7} + \left(14 a + 36\right)\cdot 47^{8} + \left(44 a + 22\right)\cdot 47^{9} + \left(19 a + 2\right)\cdot 47^{10} + \left(10 a + 37\right)\cdot 47^{11} + \left(25 a + 34\right)\cdot 47^{12} + \left(39 a + 9\right)\cdot 47^{13} + \left(13 a + 44\right)\cdot 47^{14} + \left(28 a + 3\right)\cdot 47^{15} + \left(13 a + 4\right)\cdot 47^{16} + \left(42 a + 27\right)\cdot 47^{17} + \left(35 a + 8\right)\cdot 47^{18} + \left(12 a + 25\right)\cdot 47^{19} + \left(5 a + 28\right)\cdot 47^{20} + \left(34 a + 6\right)\cdot 47^{21} + \left(19 a + 7\right)\cdot 47^{22} + \left(20 a + 45\right)\cdot 47^{23} +O\left(47^{ 24 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 8 }$

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character value
$1$$1$$()$$4$
$1$$2$$(1,8)(2,7)(3,6)(4,5)$$-4$
$4$$2$$(2,7)$$-2$
$4$$2$$(1,8)(2,7)(3,6)$$2$
$6$$2$$(2,7)(4,5)$$0$
$12$$2$$(1,3)(2,4)(5,7)(6,8)$$0$
$12$$2$$(1,4)(5,8)$$-2$
$12$$2$$(1,8)(2,3)(4,5)(6,7)$$2$
$24$$2$$(1,4)(2,7)(5,8)$$0$
$32$$3$$(1,3,2)(6,7,8)$$1$
$12$$4$$(1,3,8,6)(2,5,7,4)$$0$
$12$$4$$(1,4,8,5)$$2$
$12$$4$$(1,8)(2,6,7,3)(4,5)$$-2$
$24$$4$$(1,3)(2,5,7,4)(6,8)$$0$
$24$$4$$(1,4,8,5)(2,7)$$0$
$48$$4$$(1,4,3,2)(5,6,7,8)$$0$
$32$$6$$(1,3,2,8,6,7)$$-1$
$32$$6$$(1,3,2)(4,5)(6,7,8)$$1$
$32$$6$$(1,3,2,8,6,7)(4,5)$$-1$
$48$$8$$(1,4,3,2,8,5,6,7)$$0$
The blue line marks the conjugacy class containing complex conjugation.