Properties

Label 2.15975.4t3.a.a
Dimension $2$
Group $D_{4}$
Conductor $15975$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{4}$
Conductor: \(15975\)\(\medspace = 3^{2} \cdot 5^{2} \cdot 71 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 4.0.5671125.1
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Determinant: 1.71.2t1.a.a
Projective image: $C_2^2$
Projective field: \(\Q(\sqrt{5}, \sqrt{-71})\)

Defining polynomial

$f(x)$$=$\(x^{4} - 2 x^{3} - 6 x^{2} + 7 x + 811\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 19 }$ to precision 6.

Roots:
$r_{ 1 }$ $=$ \( 6 + 10\cdot 19 + 10\cdot 19^{2} + 18\cdot 19^{3} + 11\cdot 19^{4} + 10\cdot 19^{5} +O(19^{6})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 9 + 10\cdot 19 + 19^{2} + 9\cdot 19^{3} + 7\cdot 19^{4} + 11\cdot 19^{5} +O(19^{6})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 11 + 8\cdot 19 + 17\cdot 19^{2} + 9\cdot 19^{3} + 11\cdot 19^{4} + 7\cdot 19^{5} +O(19^{6})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 14 + 8\cdot 19 + 8\cdot 19^{2} + 7\cdot 19^{4} + 8\cdot 19^{5} +O(19^{6})\)  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.