Properties

Label 2.2e2_3_23.4t3.2c1
Dimension 2
Group $D_{4}$
Conductor $ 2^{2} \cdot 3 \cdot 23 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$2$
Group:$D_{4}$
Conductor:$276= 2^{2} \cdot 3 \cdot 23 $
Artin number field: Splitting field of $f= x^{4} + x^{2} - 6 x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $D_{4}$
Parity: Odd
Determinant: 1.2e2_3_23.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 71 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 11 + 43\cdot 71 + 8\cdot 71^{2} + 31\cdot 71^{3} + 60\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 17 + 24\cdot 71 + 25\cdot 71^{2} + 54\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 19 + 15\cdot 71 + 58\cdot 71^{2} + 58\cdot 71^{3} + 5\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 24 + 59\cdot 71 + 49\cdot 71^{2} + 51\cdot 71^{3} + 21\cdot 71^{4} +O\left(71^{ 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.