Properties

Label 6.33230963375.12t108.b.a
Dimension 6
Group $V_4^2:(S_3\times C_2)$
Conductor $ 5^{3} \cdot 643^{3}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$6$
Group:$V_4^2:(S_3\times C_2)$
Conductor:$33230963375= 5^{3} \cdot 643^{3} $
Artin number field: Splitting field of 8.4.258405625.1 defined by $f= x^{8} - 3 x^{7} + 4 x^{6} - 5 x^{5} + x^{4} + x^{3} + 8 x^{2} - 7 x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: 12T108
Parity: Odd
Determinant: 1.3215.2t1.a.a
Projective image: $C_2^2:S_4:C_2$
Projective field: Galois closure of 8.4.258405625.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 29.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $ x^{3} + 2 x + 27 $
Roots:
$r_{ 1 }$ $=$ $ 16 a^{2} + 12 a + 19 + \left(4 a^{2} + 22 a + 22\right)\cdot 29 + \left(25 a^{2} + 23 a + 8\right)\cdot 29^{2} + \left(9 a^{2} + 14 a + 9\right)\cdot 29^{3} + \left(13 a^{2} + 10 a + 23\right)\cdot 29^{4} + \left(9 a^{2} + 4 a + 9\right)\cdot 29^{5} + \left(18 a^{2} + 25 a + 20\right)\cdot 29^{6} + \left(12 a^{2} + 11\right)\cdot 29^{7} + \left(18 a^{2} + 21 a + 15\right)\cdot 29^{8} + \left(25 a^{2} + 6 a\right)\cdot 29^{9} + \left(23 a^{2} + 3\right)\cdot 29^{10} + \left(13 a^{2} + 14 a + 11\right)\cdot 29^{11} + \left(15 a^{2} + 16 a + 12\right)\cdot 29^{12} + \left(4 a^{2} + 17 a + 16\right)\cdot 29^{13} + \left(2 a^{2} + 18 a + 21\right)\cdot 29^{14} + \left(18 a^{2} + 4 a\right)\cdot 29^{15} + \left(18 a^{2} + 4 a + 6\right)\cdot 29^{16} + \left(4 a^{2} + 19 a + 22\right)\cdot 29^{17} + \left(4 a^{2} + 12 a\right)\cdot 29^{18} + \left(7 a^{2} + 15 a + 20\right)\cdot 29^{19} + \left(12 a^{2} + 25 a + 21\right)\cdot 29^{20} + \left(24 a^{2} + 8 a + 22\right)\cdot 29^{21} + \left(19 a^{2} + a + 11\right)\cdot 29^{22} + \left(26 a^{2} + 27 a + 4\right)\cdot 29^{23} + \left(12 a^{2} + 11 a + 8\right)\cdot 29^{24} + \left(27 a^{2} + 28 a + 23\right)\cdot 29^{25} + \left(21 a^{2} + 3 a + 1\right)\cdot 29^{26} + \left(22 a^{2} + 6 a + 14\right)\cdot 29^{27} + \left(5 a^{2} + 23 a + 1\right)\cdot 29^{28} +O\left(29^{ 29 }\right)$
$r_{ 2 }$ $=$ $ 25 a^{2} + 7 a + 2 + \left(5 a^{2} + 26 a + 5\right)\cdot 29 + \left(6 a^{2} + 13 a + 22\right)\cdot 29^{2} + \left(3 a^{2} + 22 a + 19\right)\cdot 29^{3} + \left(23 a^{2} + 2 a + 26\right)\cdot 29^{4} + \left(25 a^{2} + 14 a + 21\right)\cdot 29^{5} + \left(21 a^{2} + a + 5\right)\cdot 29^{6} + \left(23 a^{2} + 14 a + 7\right)\cdot 29^{7} + \left(10 a^{2} + 20 a + 5\right)\cdot 29^{8} + \left(5 a^{2} + 20 a + 12\right)\cdot 29^{9} + \left(20 a^{2} + 28 a + 17\right)\cdot 29^{10} + \left(15 a + 22\right)\cdot 29^{11} + \left(7 a^{2} + 10\right)\cdot 29^{12} + \left(20 a^{2} + 19 a + 8\right)\cdot 29^{13} + \left(10 a^{2} + 19 a + 23\right)\cdot 29^{14} + \left(28 a^{2} + 8 a + 4\right)\cdot 29^{15} + \left(25 a^{2} + 21 a + 6\right)\cdot 29^{16} + \left(26 a^{2} + 27 a + 13\right)\cdot 29^{17} + \left(6 a^{2} + 27 a + 4\right)\cdot 29^{18} + \left(3 a^{2} + 17 a + 5\right)\cdot 29^{19} + \left(23 a^{2} + 12 a + 7\right)\cdot 29^{20} + \left(13 a^{2} + 26 a + 18\right)\cdot 29^{21} + \left(26 a^{2} + 20 a + 20\right)\cdot 29^{22} + \left(17 a^{2} + 8 a + 21\right)\cdot 29^{23} + \left(4 a^{2} + 3 a + 6\right)\cdot 29^{24} + \left(12 a^{2} + 3 a + 22\right)\cdot 29^{25} + \left(24 a^{2} + a + 14\right)\cdot 29^{26} + \left(14 a^{2} + 8 a + 3\right)\cdot 29^{27} + \left(12 a^{2} + 12 a + 20\right)\cdot 29^{28} +O\left(29^{ 29 }\right)$
$r_{ 3 }$ $=$ $ 27 a + 28 + \left(9 a^{2} + 3 a + 13\right)\cdot 29 + \left(20 a^{2} + 19 a + 25\right)\cdot 29^{2} + \left(19 a^{2} + 18 a + 18\right)\cdot 29^{3} + \left(20 a + 24\right)\cdot 29^{4} + \left(27 a^{2} + 14 a + 4\right)\cdot 29^{5} + \left(22 a^{2} + 12 a\right)\cdot 29^{6} + \left(8 a^{2} + 13 a + 22\right)\cdot 29^{7} + \left(26 a^{2} + 26 a + 23\right)\cdot 29^{8} + \left(22 a^{2} + 17 a + 8\right)\cdot 29^{9} + \left(22 a^{2} + 16 a + 5\right)\cdot 29^{10} + \left(3 a^{2} + a + 2\right)\cdot 29^{11} + \left(18 a^{2} + 15 a + 14\right)\cdot 29^{12} + \left(19 a^{2} + 27 a + 7\right)\cdot 29^{13} + \left(14 a^{2} + 25\right)\cdot 29^{14} + \left(23 a^{2} + 13 a + 10\right)\cdot 29^{15} + \left(24 a^{2} + 28 a + 21\right)\cdot 29^{16} + \left(10 a^{2} + 28 a + 1\right)\cdot 29^{17} + \left(23 a^{2} + 10 a + 25\right)\cdot 29^{18} + \left(13 a^{2} + 12 a + 6\right)\cdot 29^{19} + \left(22 a^{2} + 28 a + 27\right)\cdot 29^{20} + \left(19 a^{2} + 15 a + 28\right)\cdot 29^{21} + \left(13 a^{2} + 15 a + 11\right)\cdot 29^{22} + \left(8 a^{2} + 13 a + 14\right)\cdot 29^{23} + \left(24 a^{2} + 2 a + 15\right)\cdot 29^{24} + \left(26 a^{2} + 20 a + 24\right)\cdot 29^{25} + \left(17 a^{2} + 28 a + 5\right)\cdot 29^{26} + \left(20 a^{2} + 20 a + 20\right)\cdot 29^{27} + \left(11 a^{2} + 13 a + 23\right)\cdot 29^{28} +O\left(29^{ 29 }\right)$
$r_{ 4 }$ $=$ $ 9 a^{2} + 11 + \left(2 a^{2} + 3 a + 24\right)\cdot 29 + \left(9 a^{2} + 9 a\right)\cdot 29^{2} + \left(22 a^{2} + 15 a + 3\right)\cdot 29^{3} + \left(7 a^{2} + 8 a + 5\right)\cdot 29^{4} + \left(10 a^{2} + 20 a + 21\right)\cdot 29^{5} + \left(8 a^{2} + 9 a + 9\right)\cdot 29^{6} + \left(11 a^{2} + 6 a + 25\right)\cdot 29^{7} + \left(10 a^{2} + 25 a + 21\right)\cdot 29^{8} + \left(5 a^{2} + 11 a + 4\right)\cdot 29^{9} + \left(a^{2} + 7 a + 15\right)\cdot 29^{10} + \left(13 a^{2} + 23 a + 14\right)\cdot 29^{11} + \left(3 a^{2} + 26 a + 23\right)\cdot 29^{12} + \left(12 a^{2} + 16\right)\cdot 29^{13} + \left(4 a^{2} + 18 a + 11\right)\cdot 29^{14} + \left(27 a^{2} + 12 a + 25\right)\cdot 29^{15} + \left(26 a^{2} + 19 a + 4\right)\cdot 29^{16} + \left(16 a^{2} + 23 a\right)\cdot 29^{17} + \left(27 a^{2} + 21 a + 21\right)\cdot 29^{18} + \left(13 a^{2} + 20 a + 16\right)\cdot 29^{19} + \left(6 a^{2} + 11 a + 15\right)\cdot 29^{20} + \left(5 a^{2} + 3 a + 9\right)\cdot 29^{21} + \left(27 a^{2} + 19 a + 20\right)\cdot 29^{22} + \left(a^{2} + 22 a + 5\right)\cdot 29^{23} + \left(14 a^{2} + 20 a + 21\right)\cdot 29^{24} + \left(15 a^{2} + 16 a + 28\right)\cdot 29^{25} + \left(18 a^{2} + 10 a + 25\right)\cdot 29^{26} + \left(16 a^{2} + 2 a + 14\right)\cdot 29^{27} + \left(21 a^{2} + 8 a + 17\right)\cdot 29^{28} +O\left(29^{ 29 }\right)$
$r_{ 5 }$ $=$ $ 20 a^{2} + 2 a + 16 + \left(17 a^{2} + 22 a + 25\right)\cdot 29 + \left(28 a^{2} + 26\right)\cdot 29^{2} + \left(15 a^{2} + 24 a + 13\right)\cdot 29^{3} + \left(20 a^{2} + 28 a + 12\right)\cdot 29^{4} + \left(20 a^{2} + 22 a + 25\right)\cdot 29^{5} + \left(26 a^{2} + 6 a + 14\right)\cdot 29^{6} + \left(8 a^{2} + 9 a + 12\right)\cdot 29^{7} + \left(21 a^{2} + 6 a + 7\right)\cdot 29^{8} + \left(28 a + 8\right)\cdot 29^{9} + \left(5 a^{2} + 4 a + 20\right)\cdot 29^{10} + \left(12 a^{2} + 4 a + 3\right)\cdot 29^{11} + \left(7 a^{2} + 16 a + 19\right)\cdot 29^{12} + \left(26 a^{2} + 6\right)\cdot 29^{13} + \left(9 a^{2} + 10 a + 9\right)\cdot 29^{14} + \left(7 a^{2} + 3 a + 18\right)\cdot 29^{15} + \left(6 a^{2} + 10 a + 25\right)\cdot 29^{16} + \left(a^{2} + 5 a + 17\right)\cdot 29^{17} + \left(7 a^{2} + 25 a + 22\right)\cdot 29^{18} + \left(a^{2} + 24 a + 28\right)\cdot 29^{19} + \left(17 a + 6\right)\cdot 29^{20} + \left(4 a^{2} + 9 a + 27\right)\cdot 29^{21} + \left(17 a^{2} + 23 a + 6\right)\cdot 29^{22} + \left(18 a^{2} + 21 a + 18\right)\cdot 29^{23} + \left(19 a^{2} + 5 a + 28\right)\cdot 29^{24} + \left(15 a^{2} + 21 a + 28\right)\cdot 29^{25} + \left(21 a^{2} + 18 a\right)\cdot 29^{26} + \left(20 a^{2} + 5 a + 1\right)\cdot 29^{27} + \left(24 a^{2} + 7 a + 12\right)\cdot 29^{28} +O\left(29^{ 29 }\right)$
$r_{ 6 }$ $=$ $ 14 + 21\cdot 29^{2} + 10\cdot 29^{3} + 27\cdot 29^{4} + 9\cdot 29^{5} + 11\cdot 29^{6} + 15\cdot 29^{7} + 5\cdot 29^{8} + 23\cdot 29^{9} + 3\cdot 29^{10} + 16\cdot 29^{11} + 18\cdot 29^{12} + 10\cdot 29^{13} + 2\cdot 29^{14} + 22\cdot 29^{15} + 10\cdot 29^{16} + 9\cdot 29^{17} + 10\cdot 29^{18} + 25\cdot 29^{19} + 13\cdot 29^{20} + 29^{21} + 19\cdot 29^{22} + 9\cdot 29^{23} + 11\cdot 29^{24} + 20\cdot 29^{25} + 24\cdot 29^{26} + 18\cdot 29^{27} + 20\cdot 29^{28} +O\left(29^{ 29 }\right)$
$r_{ 7 }$ $=$ $ 28 + 29 + 23\cdot 29^{3} + 5\cdot 29^{5} + 5\cdot 29^{6} + 27\cdot 29^{7} + 26\cdot 29^{8} + 26\cdot 29^{9} + 12\cdot 29^{10} + 14\cdot 29^{11} + 7\cdot 29^{12} + 14\cdot 29^{13} + 11\cdot 29^{14} + 22\cdot 29^{15} + 22\cdot 29^{16} + 9\cdot 29^{17} + 22\cdot 29^{18} + 6\cdot 29^{19} + 7\cdot 29^{20} + 20\cdot 29^{21} + 14\cdot 29^{22} + 16\cdot 29^{23} + 8\cdot 29^{24} + 24\cdot 29^{25} + 24\cdot 29^{26} + 22\cdot 29^{27} + 2\cdot 29^{28} +O\left(29^{ 29 }\right)$
$r_{ 8 }$ $=$ $ 17 a^{2} + 10 a + 1 + \left(18 a^{2} + 9 a + 22\right)\cdot 29 + \left(26 a^{2} + 20 a + 10\right)\cdot 29^{2} + \left(15 a^{2} + 20 a + 17\right)\cdot 29^{3} + \left(21 a^{2} + 15 a + 24\right)\cdot 29^{4} + \left(22 a^{2} + 10 a + 17\right)\cdot 29^{5} + \left(17 a^{2} + 2 a + 19\right)\cdot 29^{6} + \left(21 a^{2} + 14 a + 23\right)\cdot 29^{7} + \left(28 a^{2} + 16 a + 9\right)\cdot 29^{8} + \left(26 a^{2} + a + 2\right)\cdot 29^{9} + \left(13 a^{2} + 9\right)\cdot 29^{10} + \left(14 a^{2} + 28 a + 2\right)\cdot 29^{11} + \left(6 a^{2} + 11 a + 10\right)\cdot 29^{12} + \left(4 a^{2} + 21 a + 6\right)\cdot 29^{13} + \left(16 a^{2} + 19 a + 11\right)\cdot 29^{14} + \left(11 a^{2} + 15 a + 11\right)\cdot 29^{15} + \left(13 a^{2} + 3 a + 18\right)\cdot 29^{16} + \left(26 a^{2} + 11 a + 12\right)\cdot 29^{17} + \left(17 a^{2} + 17 a + 9\right)\cdot 29^{18} + \left(18 a^{2} + 24 a + 6\right)\cdot 29^{19} + \left(22 a^{2} + 19 a + 16\right)\cdot 29^{20} + \left(19 a^{2} + 22 a + 16\right)\cdot 29^{21} + \left(11 a^{2} + 6 a + 10\right)\cdot 29^{22} + \left(13 a^{2} + 22 a + 25\right)\cdot 29^{23} + \left(11 a^{2} + 13 a + 15\right)\cdot 29^{24} + \left(18 a^{2} + 26 a + 1\right)\cdot 29^{25} + \left(11 a^{2} + 23 a + 17\right)\cdot 29^{26} + \left(20 a^{2} + 14 a + 20\right)\cdot 29^{27} + \left(10 a^{2} + 22 a + 17\right)\cdot 29^{28} +O\left(29^{ 29 }\right)$

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

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

Character values on conjugacy classes

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