Properties

Label 2.2912.6t3.b.a
Dimension 22
Group D6D_{6}
Conductor 29122912
Root number 11
Indicator 11

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Basic invariants

Dimension: 22
Group: D6D_{6}
Conductor: 29122912=25713\medspace = 2^{5} \cdot 7 \cdot 13
Frobenius-Schur indicator: 11
Root number: 11
Artin stem field: Galois closure of 6.0.16959488.2
Galois orbit size: 11
Smallest permutation container: D6D_{6}
Parity: odd
Determinant: 1.728.2t1.b.a
Projective image: S3S_3
Projective stem field: Galois closure of 3.1.728.1

Defining polynomial

f(x)f(x)== x6+10x4+25x2+32 x^{6} + 10x^{4} + 25x^{2} + 32 Copy content Toggle raw display .

The roots of ff are computed in an extension of Q17\Q_{ 17 } to precision 8.

Minimal polynomial of a generator aa of KK over Q17\mathbb{Q}_{ 17 }: x2+16x+3 x^{2} + 16x + 3 Copy content Toggle raw display

Roots:
r1r_{ 1 } == 1+1217+3172+4173+11174+15175+3176+8177+O(178) 1 + 12\cdot 17 + 3\cdot 17^{2} + 4\cdot 17^{3} + 11\cdot 17^{4} + 15\cdot 17^{5} + 3\cdot 17^{6} + 8\cdot 17^{7} +O(17^{8}) Copy content Toggle raw display
r2r_{ 2 } == 4a+7+(16a+8)17+(11a+12)172+(10a+2)173+(a+10)174+8175+(5a+16)176+14177+O(178) 4 a + 7 + \left(16 a + 8\right)\cdot 17 + \left(11 a + 12\right)\cdot 17^{2} + \left(10 a + 2\right)\cdot 17^{3} + \left(a + 10\right)\cdot 17^{4} + 8\cdot 17^{5} + \left(5 a + 16\right)\cdot 17^{6} + 14\cdot 17^{7} +O(17^{8}) Copy content Toggle raw display
r3r_{ 3 } == 4a+6+(16a+13)17+(11a+8)172+(10a+15)173+(a+15)174+9175+(5a+12)176+6177+O(178) 4 a + 6 + \left(16 a + 13\right)\cdot 17 + \left(11 a + 8\right)\cdot 17^{2} + \left(10 a + 15\right)\cdot 17^{3} + \left(a + 15\right)\cdot 17^{4} + 9\cdot 17^{5} + \left(5 a + 12\right)\cdot 17^{6} + 6\cdot 17^{7} +O(17^{8}) Copy content Toggle raw display
r4r_{ 4 } == 16+417+13172+12173+5174+175+13176+8177+O(178) 16 + 4\cdot 17 + 13\cdot 17^{2} + 12\cdot 17^{3} + 5\cdot 17^{4} + 17^{5} + 13\cdot 17^{6} + 8\cdot 17^{7} +O(17^{8}) Copy content Toggle raw display
r5r_{ 5 } == 13a+10+817+(5a+4)172+(6a+14)173+(15a+6)174+(16a+8)175+11a176+(16a+2)177+O(178) 13 a + 10 + 8\cdot 17 + \left(5 a + 4\right)\cdot 17^{2} + \left(6 a + 14\right)\cdot 17^{3} + \left(15 a + 6\right)\cdot 17^{4} + \left(16 a + 8\right)\cdot 17^{5} + 11 a\cdot 17^{6} + \left(16 a + 2\right)\cdot 17^{7} +O(17^{8}) Copy content Toggle raw display
r6r_{ 6 } == 13a+11+317+(5a+8)172+(6a+1)173+(15a+1)174+(16a+7)175+(11a+4)176+(16a+10)177+O(178) 13 a + 11 + 3\cdot 17 + \left(5 a + 8\right)\cdot 17^{2} + \left(6 a + 1\right)\cdot 17^{3} + \left(15 a + 1\right)\cdot 17^{4} + \left(16 a + 7\right)\cdot 17^{5} + \left(11 a + 4\right)\cdot 17^{6} + \left(16 a + 10\right)\cdot 17^{7} +O(17^{8}) Copy content Toggle raw display

Generators of the action on the roots r1,,r6r_1, \ldots, r_{ 6 }

Cycle notation
(2,6)(3,5)(2,6)(3,5)
(1,2)(3,6)(4,5)(1,2)(3,6)(4,5)

Character values on conjugacy classes

SizeOrderAction on r1,,r6r_1, \ldots, r_{ 6 } Character valueComplex conjugation
1111()()22
1122(1,4)(2,5)(3,6)(1,4)(2,5)(3,6)2-2
3322(1,2)(3,6)(4,5)(1,2)(3,6)(4,5)00
3322(1,3)(4,6)(1,3)(4,6)00
2233(1,5,3)(2,6,4)(1,5,3)(2,6,4)1-1
2266(1,6,5,4,3,2)(1,6,5,4,3,2)11