Properties

Label 2.723.4t3.a
Dimension 22
Group D4D_{4}
Conductor 723723
Indicator 11

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Basic invariants

Dimension:22
Group:D4D_{4}
Conductor:723723=3241\medspace = 3 \cdot 241
Frobenius-Schur indicator: 11
Root number: 11
Artin number field: Galois closure of 4.0.2169.1
Galois orbit size: 11
Smallest permutation container: D4D_{4}
Parity: odd
Projective image: C22C_2^2
Projective field: Galois closure of Q(3,241)\Q(\sqrt{-3}, \sqrt{241})

Galois action

Roots of defining polynomial

The roots of ff are computed in Q181\Q_{ 181 } to precision 5.
Roots:
r1r_{ 1 } == 23+114181+1711812+751813+501814+O(1815) 23 + 114\cdot 181 + 171\cdot 181^{2} + 75\cdot 181^{3} + 50\cdot 181^{4} +O(181^{5}) Copy content Toggle raw display
r2r_{ 2 } == 110+68181+931812+1081813+201814+O(1815) 110 + 68\cdot 181 + 93\cdot 181^{2} + 108\cdot 181^{3} + 20\cdot 181^{4} +O(181^{5}) Copy content Toggle raw display
r3r_{ 3 } == 111+123181+1611812+1161813+1271814+O(1815) 111 + 123\cdot 181 + 161\cdot 181^{2} + 116\cdot 181^{3} + 127\cdot 181^{4} +O(181^{5}) Copy content Toggle raw display
r4r_{ 4 } == 119+55181+1161812+601813+1631814+O(1815) 119 + 55\cdot 181 + 116\cdot 181^{2} + 60\cdot 181^{3} + 163\cdot 181^{4} +O(181^{5}) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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