Properties

Label 2.2e2_17e2.4t3.2c1
Dimension 2
Group $D_{4}$
Conductor $ 2^{2} \cdot 17^{2}$
Root number 1
Frobenius-Schur indicator 1

Related objects

Learn more about

Basic invariants

Dimension:$2$
Group:$D_{4}$
Conductor:$1156= 2^{2} \cdot 17^{2} $
Artin number field: Splitting field of $f= x^{4} - 2 x^{3} + 10 x^{2} + 8 x + 16 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $D_{4}$
Parity: Odd
Determinant: 1.2e2.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 149 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 27 + 20\cdot 149 + 54\cdot 149^{2} + 3\cdot 149^{3} + 130\cdot 149^{4} +O\left(149^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 44 + 83\cdot 149 + 76\cdot 149^{2} + 130\cdot 149^{3} + 58\cdot 149^{4} +O\left(149^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 113 + 2\cdot 149 + 127\cdot 149^{2} + 108\cdot 149^{3} + 36\cdot 149^{4} +O\left(149^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 116 + 42\cdot 149 + 40\cdot 149^{2} + 55\cdot 149^{3} + 72\cdot 149^{4} +O\left(149^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 4 }$

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 4 }$ Character value
$1$$1$$()$$2$
$1$$2$$(1,2)(3,4)$$-2$
$2$$2$$(1,3)(2,4)$$0$
$2$$2$$(1,2)$$0$
$2$$4$$(1,4,2,3)$$0$
The blue line marks the conjugacy class containing complex conjugation.