Properties

Label 2.2e4_3e2_5e2.4t3.1c1
Dimension 2
Group $D_{4}$
Conductor $ 2^{4} \cdot 3^{2} \cdot 5^{2}$
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$2$
Group:$D_{4}$
Conductor:$3600= 2^{4} \cdot 3^{2} \cdot 5^{2} $
Artin number field: Splitting field of $f= x^{4} - 45 $ 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 $\Q_{ 29 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 2 + 10\cdot 29 + 18\cdot 29^{2} + 16\cdot 29^{3} + 10\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 5 + 22\cdot 29 + 6\cdot 29^{2} + 7\cdot 29^{3} + 25\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 24 + 6\cdot 29 + 22\cdot 29^{2} + 21\cdot 29^{3} + 3\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 27 + 18\cdot 29 + 10\cdot 29^{2} + 12\cdot 29^{3} + 18\cdot 29^{4} +O\left(29^{ 5 }\right)$

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

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

Character values on conjugacy classes

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