Properties

Label 2.1575.4t3.a.a
Dimension $2$
Group $D_{4}$
Conductor $1575$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{4}$
Conductor: \(1575\)\(\medspace = 3^{2} \cdot 5^{2} \cdot 7 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 4.0.4725.1
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Determinant: 1.7.2t1.a.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{-3}, \sqrt{-7})\)

Defining polynomial

$f(x)$$=$ \( x^{4} - x^{3} + 12x^{2} - 4x + 31 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 37 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 14 + 25\cdot 37 + 6\cdot 37^{2} + 36\cdot 37^{3} + 30\cdot 37^{4} +O(37^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 29 + 30\cdot 37 + 21\cdot 37^{2} + 18\cdot 37^{3} + 34\cdot 37^{4} +O(37^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 34 + 32\cdot 37 + 19\cdot 37^{2} + 27\cdot 37^{3} + 6\cdot 37^{4} +O(37^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 35 + 21\cdot 37 + 25\cdot 37^{2} + 28\cdot 37^{3} + 37^{4} +O(37^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 4 }$ Character value
$1$$1$$()$$2$
$1$$2$$(1,3)(2,4)$$-2$
$2$$2$$(1,2)(3,4)$$0$
$2$$2$$(1,3)$$0$
$2$$4$$(1,4,3,2)$$0$

The blue line marks the conjugacy class containing complex conjugation.