Properties

Label 2.740.4t3.c.a
Dimension 22
Group D4D_{4}
Conductor 740740
Root number 11
Indicator 11

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Basic invariants

Dimension: 22
Group: D4D_{4}
Conductor: 740740=22537\medspace = 2^{2} \cdot 5 \cdot 37
Frobenius-Schur indicator: 11
Root number: 11
Artin stem field: Galois closure of 4.0.2960.1
Galois orbit size: 11
Smallest permutation container: D4D_{4}
Parity: odd
Determinant: 1.740.2t1.a.a
Projective image: C22C_2^2
Projective field: Galois closure of Q(i,185)\Q(i, \sqrt{185})

Defining polynomial

f(x)f(x)== x42x35x2+6x+10 x^{4} - 2x^{3} - 5x^{2} + 6x + 10 Copy content Toggle raw display .

The roots of ff are computed in Q41\Q_{ 41 } to precision 5.

Roots:
r1r_{ 1 } == 4+1341+39412+15413+37414+O(415) 4 + 13\cdot 41 + 39\cdot 41^{2} + 15\cdot 41^{3} + 37\cdot 41^{4} +O(41^{5}) Copy content Toggle raw display
r2r_{ 2 } == 16+2241+33412+20413+20414+O(415) 16 + 22\cdot 41 + 33\cdot 41^{2} + 20\cdot 41^{3} + 20\cdot 41^{4} +O(41^{5}) Copy content Toggle raw display
r3r_{ 3 } == 26+1841+7412+20413+20414+O(415) 26 + 18\cdot 41 + 7\cdot 41^{2} + 20\cdot 41^{3} + 20\cdot 41^{4} +O(41^{5}) Copy content Toggle raw display
r4r_{ 4 } == 38+2741+412+25413+3414+O(415) 38 + 27\cdot 41 + 41^{2} + 25\cdot 41^{3} + 3\cdot 41^{4} +O(41^{5}) Copy content Toggle raw display

Generators of the action on the roots r1,,r4r_1, \ldots, r_{ 4 }

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

Character values on conjugacy classes

SizeOrderAction on r1,,r4r_1, \ldots, r_{ 4 } Character valueComplex conjugation
1111()()22
1122(1,4)(2,3)(1,4)(2,3)2-2
2222(1,2)(3,4)(1,2)(3,4)00
2222(1,4)(1,4)00
2244(1,3,4,2)(1,3,4,2)00