Properties

Label 2.279.4t3.b.a
Dimension 22
Group D4D_{4}
Conductor 279279
Root number 11
Indicator 11

Related objects

Downloads

Learn more

Basic invariants

Dimension: 22
Group: D4D_{4}
Conductor: 279279=3231\medspace = 3^{2} \cdot 31
Frobenius-Schur indicator: 11
Root number: 11
Artin stem field: Galois closure of 4.0.837.1
Galois orbit size: 11
Smallest permutation container: D4D_{4}
Parity: odd
Determinant: 1.31.2t1.a.a
Projective image: C22C_2^2
Projective field: Galois closure of Q(3,31)\Q(\sqrt{-3}, \sqrt{-31})

Defining polynomial

f(x)f(x)== x4x33x2x+7 x^{4} - x^{3} - 3x^{2} - x + 7 Copy content Toggle raw display .

The roots of ff are computed in Q7\Q_{ 7 } to precision 5.

Roots:
r1r_{ 1 } == 7+472+273+274+O(75) 7 + 4\cdot 7^{2} + 2\cdot 7^{3} + 2\cdot 7^{4} +O(7^{5}) Copy content Toggle raw display
r2r_{ 2 } == 4+472+574+O(75) 4 + 4\cdot 7^{2} + 5\cdot 7^{4} +O(7^{5}) Copy content Toggle raw display
r3r_{ 3 } == 5+7+372+474+O(75) 5 + 7 + 3\cdot 7^{2} + 4\cdot 7^{4} +O(7^{5}) Copy content Toggle raw display
r4r_{ 4 } == 6+37+272+373+274+O(75) 6 + 3\cdot 7 + 2\cdot 7^{2} + 3\cdot 7^{3} + 2\cdot 7^{4} +O(7^{5}) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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