Properties

Label 2.1469.4t3.b.a
Dimension $2$
Group $D_{4}$
Conductor $1469$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{4}$
Conductor: \(1469\)\(\medspace = 13 \cdot 113 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 4.0.165997.1
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: even
Determinant: 1.1469.2t1.a.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{13}, \sqrt{113})\)

Defining polynomial

$f(x)$$=$ \( x^{4} - 2x^{3} + 18x^{2} - 17x + 44 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 53 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 17 + 38\cdot 53^{2} + 25\cdot 53^{3} + 49\cdot 53^{4} +O(53^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 23 + 35\cdot 53 + 42\cdot 53^{2} + 4\cdot 53^{3} + 5\cdot 53^{4} +O(53^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 31 + 17\cdot 53 + 10\cdot 53^{2} + 48\cdot 53^{3} + 47\cdot 53^{4} +O(53^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 37 + 52\cdot 53 + 14\cdot 53^{2} + 27\cdot 53^{3} + 3\cdot 53^{4} +O(53^{5})\) Copy content Toggle raw display

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

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

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.