Properties

Label 2.276.4t3.e.a
Dimension $2$
Group $D_{4}$
Conductor $276$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{4}$
Conductor: \(276\)\(\medspace = 2^{2} \cdot 3 \cdot 23 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 4.2.3312.1
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Determinant: 1.276.2t1.a.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{3}, \sqrt{-23})\)

Defining polynomial

$f(x)$$=$ \( x^{4} - 2x^{3} - x^{2} + 2x - 2 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 47 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 19 + 6\cdot 47 + 6\cdot 47^{2} + 14\cdot 47^{3} + 40\cdot 47^{4} +O(47^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 23 + 26\cdot 47 + 12\cdot 47^{2} + 12\cdot 47^{3} + 23\cdot 47^{4} +O(47^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 25 + 20\cdot 47 + 34\cdot 47^{2} + 34\cdot 47^{3} + 23\cdot 47^{4} +O(47^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 29 + 40\cdot 47 + 40\cdot 47^{2} + 32\cdot 47^{3} + 6\cdot 47^{4} +O(47^{5})\) Copy content Toggle raw display

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.