Properties

Label 2.1872.4t3.b
Dimension 22
Group D4D_{4}
Conductor 18721872
Indicator 11

Related objects

Downloads

Learn more

Basic invariants

Dimension:22
Group:D4D_{4}
Conductor:18721872=243213\medspace = 2^{4} \cdot 3^{2} \cdot 13
Frobenius-Schur indicator: 11
Root number: 11
Artin number field: Galois closure of 4.0.5616.1
Galois orbit size: 11
Smallest permutation container: D4D_{4}
Parity: odd
Projective image: C22C_2^2
Projective field: Galois closure of Q(3,13)\Q(\sqrt{-3}, \sqrt{-13})

Galois action

Roots of defining polynomial

The roots of ff are computed in Q7\Q_{ 7 } to precision 7.
Roots:
r1r_{ 1 } == 1+37+672+73+675+76+O(77) 1 + 3\cdot 7 + 6\cdot 7^{2} + 7^{3} + 6\cdot 7^{5} + 7^{6} +O(7^{7}) Copy content Toggle raw display
r2r_{ 2 } == 2+47+472+573+75+O(77) 2 + 4\cdot 7 + 4\cdot 7^{2} + 5\cdot 7^{3} + 7^{5} +O(7^{7}) Copy content Toggle raw display
r3r_{ 3 } == 5+27+272+73+674+575+676+O(77) 5 + 2\cdot 7 + 2\cdot 7^{2} + 7^{3} + 6\cdot 7^{4} + 5\cdot 7^{5} + 6\cdot 7^{6} +O(7^{7}) Copy content Toggle raw display
r4r_{ 4 } == 6+37+573+674+576+O(77) 6 + 3\cdot 7 + 5\cdot 7^{3} + 6\cdot 7^{4} + 5\cdot 7^{6} +O(7^{7}) 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)
(2,3)(2,3)

Character values on conjugacy classes

SizeOrderAction on r1,,r4r_1, \ldots, r_{ 4 } Character values
c1c1
11 11 ()() 22
11 22 (1,4)(2,3)(1,4)(2,3) 2-2
22 22 (1,2)(3,4)(1,2)(3,4) 00
22 22 (1,4)(1,4) 00
22 44 (1,3,4,2)(1,3,4,2) 00
The blue line marks the conjugacy class containing complex conjugation.