Properties

Label 2.224.4t3.c.a
Dimension 2
Group $D_{4}$
Conductor $ 2^{5} \cdot 7 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$2$
Group:$D_{4}$
Conductor:$224= 2^{5} \cdot 7 $
Artin number field: Splitting field of 4.2.1792.1 defined by $f= x^{4} - 2 x^{2} - 4 x - 2 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $D_{4}$
Parity: Odd
Determinant: 1.56.2t1.b.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{2}, \sqrt{-7})\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 71 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 27 + 12\cdot 71 + 13\cdot 71^{2} + 68\cdot 71^{3} + 20\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 32 + 64\cdot 71 + 17\cdot 71^{2} + 24\cdot 71^{3} + 62\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 36 + 60\cdot 71 + 69\cdot 71^{2} + 48\cdot 71^{3} + 53\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 47 + 4\cdot 71 + 41\cdot 71^{2} + 5\cdot 71^{4} +O\left(71^{ 5 }\right)$

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

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

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.